Xrfgtjtk

No, it isn't a contradiction under US law. This has been tested in the Federal Courts, see for example O'Hair v. Blumenthal, and Aronow v. United States. The basic reasoning is summarized in this paragraph from the Anonow case:

It is quite obvious that the national motto and the slogan on coinage and currency "In God We Trust" has nothing whatsoever to do with the establishment of religion. Its use is of a patriotic or ceremonial character and bears no true resemblance to a governmental sponsorship of a religious exercise.

People can certainly disagree based on opinion, but in context of politics and law, the mention of God isn't establishment of a state religion in the U.S.

This is a rather principialist question, and politics is a much more pragmatic field.

Philosophically, yes, a State should not acknowledge a god, there are religions without gods, agnosticism and atheism. As the State should not endorses any group, the god question should be absent from the State sphere.

Pragmatically, four in five Americans are from a abraamic religion that share a root concept of god, they fell represented and will defend the use of this symbology, even hurting the neutrality of the State. No politician would campaign against it, as there is no practical gain and a huge practical loss for them. Unless non christians/jews/mulisms, that are prejudiced by this "state monotheism", campaign actively against it, nothing would change.

Disclaimer: I'm not that versed in history, so the following might have factual mistakes.

To understand the source of the american separation of church and state, you'll have to take a dive into history.

Between the 16th and 18th century, Europe was in a state of religious turmoil. The ruler decided which religion his subjects would have to follow (Cuius regio, eius religio (thanks @molnarm)), so if you were born under a catholic ruler and your village was then conquered by a protestant, you'd have to convert or be persecuted, banished or worse. Just paying lip service to a religion could also have dire consequences. If you then got re-conquered, you'd have to convert again. Due to pretty much constant wars, such a situation wasn't too uncommon.

America became a safe haven for people where they could live without anyone forcing their religion on them, which is why many victims of religious persecution and bloody wars chose to migrate west. The thought of "We don't want anyone to tell us which religion we must have" became ingrained into minds of the american population, but it was still assumed that everyone was a christian. When the USA achieved independence, the founders decided to formalize it so that a situation like in Europe would never happen on American soil.

The american separation of church and state is not about separating religion from politics, it's about separating the citizens from the religion of the ruler. The USA was still founded as a pretty much christian nation, but the founders wanted to guarantee that no citizen would be forced into a specific denomination on the whim of a ruler.

In the general case, yes, acknowledging a god as a state means that you discriminate against polytheist churches and atheist belief systems. You connect the state to all monotheistic churches. However, many European and American countries were formed or are based on countries formed during a time when it was assumed that everyone was some sort of christian and there wasn't enough political pressure for stricter separation and so the connection between many American and European states and Christianity in general still exists.

As a side note, the oath that a president has to swear doesn't have to include the bible. The president can choose what he swears on. It's just that most are Christians and decide to swear on the bible.

"Separation of church and state" is not actually required by the US Constitution.

Wikipedia says that the phrase "separation of church and state" was first used by Thomas Jefferson in this letter in 1802, when he was President:

Believing with you that religion is a matter which lies solely between
Man & his God, that he owes account to none other for his faith or his
worship, that the legitimate powers of government reach actions only,
& not opinions, I contemplate with sovereign reverence that act of the
whole American people which declared that their legislature should
"make no law respecting an establishment of religion, or prohibiting
the free exercise thereof," thus building a wall of separation between
Church & State.

Jefferson's use of the phrase indicates that it is the effect of the First Amendment restrictions on the government, but it is not the literal language. Justice Hugo Black wrote that the phrase expresses the intent of the literal language of the amendment.

This matters a great deal. A court case concerning the First Amendment should look at whether a particular issue involves an establishment of religion, or a prohibition of the free exercise of religion. But it is actually not relevant whether a particular issue involves, or violates, separation of church and state - because there is no such requirement in the Constitution!

Usually the distinction is minor, but it can be important. One example, that OP mentioned, is the use of "In God We Trust" on US bills and coins. It's pretty obvious that this is not compatible with the idea of separation of church and state. But that doesn't actually matter. It does not establish religion, and therefore it is permissible according to the Supreme Court.

There is considerable legal debate about this point, though, as described in the previously linked paper. The divide along liberal/conservative lines is basically around whether the First Amendment requires the state to be secular, or simply requires that it not give advantage or disadvantage.

Criticism of the modern Court’s separationist approach has existed since the 1940s, but gained momentum as a result of the resurgence of conservatism during the 1980s and the appointment of constitutional conservatives to the Supreme Court. Today, it is not uncommon for religious, legal, and cultural conservatives to criticize the concept of church-state separation. Critics charge that a separationist perspective imposes a regime of secularism, one that is not neutral toward religious matters but that privatizes and marginalizes religion. Yale law professor Stephen L. Carter argued that the separationism promoted “a culture of disbelief,” while Catholic theologian Richard John Neuhaus claimed that it created a religiously “naked public square.

I will try to explain as I was taught, so please bear with me.

Separation of church and state is just that: The state should hold no powers over any church.

Keep in mind that, while not a new concept back then, it was an unusual one. Many "states" like Spain, England, France, and so on were actually going through quite a bit of religious turmoil.

It was illegal, and sometimes even a capital crime to believe differently than the government. Monarchs in those days could not even take the throne without the "permission" of the pope. In fact, monarchs are "imbued by God with the power to rule". They are literally better then the rest of us because God said so. At least that's how the logic went back then. (Some of that was starting to change, but it was still recent).

The separation of church and state was to prevent that.

The actual line is:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof.

Which loosely means that congress can not pass a law "messin' with churches." It does not, in any way, provide for a secular country. Its purpose was exactly the opposite. The founding fathers wanted US citizens to practice whatever religion they wanted to, with no restrictions.

As for acknowledging God, there is no prohibition against it. It is in fact encouraged.

Before addressing this, a couple of points in your premise are flawed -

"...so help me God" is not part of the official swearing in. Someone can say that if they want, but it is not a required part of the official ceremony.

Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation:—“I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States.”

US Consitution - Article II, Section 1 - 8

The same goes with "God bless America" - that's a personal statement.

Now, on to your statement -

"in God We Trust" - of course it is a very specific religious statement, and intended to be a specifically Christian one. Yes, federal courts have ruled that it's some sort of generic, non-religious statement, but that's simply because SCOTUS decided they did not want to make what would be a spectacularly unpopular ruling to bar it, and basically started at their desired ruling and worked backwards from there. It was never part of the official lexicon of the US government at the time of the founding of the nation, and has only appeared and been added in response to upswings in religious sentiment.

During the Red Scare hysteria of post-WWII, early Cold War era, everyone was eager to demonstrate how non-communist they were, and the thing many fear-mongering conservatives loved to point to was the official atheist stance of the communist governments. That was added to show how non-commie and God-fearing we were, and by God-fearing they did not mean any kind of non-Protestant god. That's when it was officially, by joint Congressional proclamation, added to all currency.

“Nothing can be more certain than that our country was founded in a spiritual atmosphere and with a firm trust in God,” Bennett proclaimed on the House Floor. “While the sentiment of trust in God is universal and timeless, these particular four words ‘In God We Trust’ are indigenous to our country.” Furthermore, Bennett invoked the cold war struggle in arguing for the measure. “In these days when imperialistic and materialistic communism seeks to attack and destroy freedom, we should continually look for ways to strengthen the foundations of our freedom,” he said. Adding “In God We Trust” to currency, Bennett believed, would “serve as a constant reminder” that the nation’s political and economic fortunes were tied to its spiritual faith.

US House of Representatives History Art and Archives Historical Highlights: "In God We Trust"

Charles E Bennett, from Florida, was the member of the House of Representatives that introduced the joint resolution to enshrine "In God We Trust" as the national motto of the USA.

“In God We Trust” was first added to U.S. coins during the beginning of the Civil War, when religious sentiment was on an upswing and concerned Americans wanted the world to know what their country stood for. Many wrote to Secretary of the Treasury Salmon P. Chase on the matter, and he agreed with their arguments. Congress passed his act requesting the addition of “In God We Trust,” adapted from a lesser-known verse of Francis Scott Key’s “Star-Spangled Banner,” and the first two-cent coin with the phrase was minted in 1864.

By the turn of the century, however, the war’s memory had faded; President Teddy Roosevelt considered the mingling of God and Mammon to be vulgar, and he ordered the phrase removed from newly designed gold coins in 1907. A public outcry forced Congress to backtrack. By the mid-1950s, the concern with piety in Washington had apparently deepened; in 1955 Congress ordered the same phrase to appear on all paper currency.

....But as TIME wrote in that ’91 story, the banality of the phrases may not be worth the fight as a symbol of separating church from state. “Today even ardent separationists seem to agree with retired Supreme Court Justice William Brennan, who wrote in 1983 that slogans such as ‘In God We Trust’ have ‘lost any true religious significance.'”

Time Magazine: How 'In God We Trust' Got on the Currency in the First Place

Note that the examples you cite do not specify which God is being referred to. If the government made reference to Jesus or the ten commandments, that would be different.

That vagueness seems to be accepted as good enough both politically and legally. It's too vague to be classed as establishing a religion.

Atheists and some non-Abrahamic religions are unfortunately left out by this. That's a political consequence of those groups being a tiny minority in the US currently and even smaller historically. (However, that is slowly changing and the situation may look different in 50 years.)

And many people in those groups simply ignore such vague references to God. Many people say "Oh my God" in a purely secular manner and interpret these official references similarly. So even amongst that small group, there is not much political will to change this.

It is a (arguably slight) violation of separation of state and religion, as it implies the existence of a god and also (arguably) of a single god. However, that doesn't specifically imply support for a single religion, so it would only discriminate against atheists and potentially polytheists.
Note that it doesn't force or endorse a single religion and thus, doesn't, by itself, restrict an individual's choice of religion. It sure might offend though.

However, it is not a violation of separation of state and church, as it does not identify any particular church, imply support or preference for any church or support any particular church with means of the state.

Now this argument is general in nature, it is a different question whether the imprint violates any US law that might enshrine some form of separation of state and religion/church. There are many countries that are in principle secular states, aim to be, have laws in that direction and/or are generally considered as such, but do have some lawful entanglements with particular religions/churches, typically the (historical) majority denominations.

The founding fathers were largely theists and largely Christian, but from various different sects of Christianity. A substantial minority like Ben Franklin were deists.

Since they couldn't all agree which type of Christianity or even whether Christianity was preferable to a more generic deism, they put the separation of church and state provisos in to prevent one sect from taking over and banning all the others by government fiat.

There was not likely any serious intent that the idea of God would be abandoned, as evidenced by the official language you mention in your question.

Further more two of the largest groups to settle America (and have the first successful settlement) were the Puritans and the Quakers, both of which came here fleeing religious persecution. The idea of religious tolerance was probably a little more appealing in early America than most places.

"Isn't acknowledging the existence of God, as a state, a contradiction of the separation of Church and State?"

Your question contains the answer.

God and Religion are not one and the same - they are two entirely different things. Many religions are about God. On the other hand, God isn't about a religion, or 'church'.

There's a separation of church and state.

Nowhere is it written that there be a separation of God and state.

QED.

Without addressing your question per se, let's examine a parallel issue:

Does separation of church and state require the US government to promote atheism?

I think the answer is very clearly "No, that is not required by separation of church and state." The two extremes of building religion and destroying religion are addressed in turn by the first two clauses in the First Amendment:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof...

A law with respect to the establishment of religion, such as the Church of England or the Church of Denmark, would be an unconstitutional law. Likewise, a law prohibiting the free exercise of religion, such as the laws which regulate the order in which akharas enter the water during Kumbh Mela, would be unconstitutional.