The closest America comes is in the Declaration of Independence:
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.
"Their Creator" is about as non-denominational as it gets. It's not like it's saying "endowed by Jesus God." It's more of a generic "men are born with these rights" statement than any statement of religion.
Yet the Christian Fundamentalists will point to that and say "See? That means God and clearly it refers to our God so that means it's a Christian country!"
Oh, and a big reason why the US is secular? We broke off from England where the king was the head of the Church. If you weren't part of the Church of England, you were persecuted against. The founding fathers were trying to avoid that in the US.