Let us just say hypothetically, for the sake of the exercise, that The United States would have been set up by the Puratins from NE, or the Quakers of PA, - Virginia was of it's own brand-or the whatever the hell they were in those original thirteen mini theocracies.
If there was ever a miracle it was that men of reason (think Thomas Paine) could transcend all of that and come up with a founding document guaranteeing all of our personal beliefs.
And, after creating that document they had the foresight to make an ammendment. The first one. It wasn't the first one by accident.
Having said that, consider this. What would it be like if one religion had gained control and the Bible, and their interpretaion of that Bible would have become our founding document?