The Constitution of the U.S. is not in any sense based on religion or, in particular, Christianity. By attaching the label Christian to the 50+ Founding Fathers doesn't begin to describe them. They are better described as men of the enlightenment, who were motivated by science, intellectualism and reason. They were inspired more by Locke than Leviticus.

The pre-Amendment Constitution only mentions religion once, and that is to mandate that there will never be a religious test for holding office in the new government. Madison and others were skeptical about religion entering politics and feared that majoritarian sects would seek to manipulate federal offices to exclude others and dictate their religion on others.