I'm perfectly willing to believe that the USA's institutional structures have a somewhat Protestant cast; indeed, it makes a lot of sense. But there's a pretty big distinction between acknowledging Protestant influence on our institutions and granting Protestant Christianity a privileged position within them; if we're going to call the United States a Christian country on that account, then we should equally be prepared to call it a Deist country, an Iroquois country, and an Athenian country.

My understanding of the federal system grants a certain amount of weight to your argument regarding the government of the particular states, albeit with caveats stemming from the supremacy of the equal protection clause. On the federal level, however, you can't get much clearer than Washington's comments on the subject in the Treaty of Tripoli (or Madison and Hamilton's in the Federalist Papers, though those don't have any legal force).