I agree with the basic arguments made in each of these posts, but I would add that it is not exactly true that separation of church and state accompanied the end of the Revolution. The First Amendment only guaranteed that there would be no federal establishment--established churches persisted for some time after the Revolution in some states. Massachusetts, for example, did not rid itself of the established Congregational Church until the 1830s. I offer this only as an interesting fact--I agree with the larger point that the Revolution weakened established churches as sources of political authority.
Beyond what has already been written, I think many people interpreted (and continue to interpret) the Revolution as further evidence that the United States is somehow exceptional, favored by divine Providence.