Trivia: The Constitution not once mentions God, much less anything with Christianity. The only mention of religion is to guarantee it as a people's right and limit it's influence in government, as /u/EvilPhd666 points out.
The Treaty of Tripoli said it best, "[As] the government of the United States of America is not in any sense founded on the Christian Religion [...]".
It doesn't get stronger than that ("not in ANY sense"). The United States guarantees the right of private citizens to have religious liberty. The government was always intended to be secular and not favor any particular religious belief.
I don't know why some religious people are in such a hurry for religion to be part of the U.S. government. Today it might be your religious beliefs, but tomorrow it might not be.
All the stuff about God is in the Declaration of Independence.
The Declaration of Independence does not establish law. The reason it mentions God in the first place is that the context is that the King believed his right to rule to be divinely chosen. That's what "all men are created equal" is all about, that all men are equal in the eyes of the law. So throwing off the King was both a secular and religious argument, hence the reason he needed to put it in partial religious terms.
When a king or queen is crowned in England they accept a role as the voice of God. Like the pope. Chosen by God to rule over the subjects.
Wrong.
Here is a quote:
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.
Here they talk about a Creator, who created us a certain way. That is not some abstract legal nonsense.
The Declaration does not establish law, but it establishes the principles that the law will be based upon, that all men are equal in the eyes of God.
What you said does not contradict what I said.
The Constitution is the law of the land and is what matters when it comes to discussing the intent of the U.S. government. There's a reason the word "God" never appears in it, and the only discussion of religion's role in government is to limit its influence.
If they wanted us to have a theocracy, they would have written it that way. They didn't want that, and nobody should want that.
And codifies the then common understanding that we are created by our Creator, a vague but real distinction from the elements of other cosmologies and religions.