It might be useful to look at The Virginia Declaration of Rights, which existed before the Revolution, and which is regarded as having been the source and inspiration for much of what followed. Section 16 discusses religion, and says that it must be voluntary, and "it is the mutual duty of all to practise Christian forbearance, love, and charity toward each other." Quite non-sectarian, and non-dogmatic, but nevertheless inspired by the Christian religion.
Thank you so much PatrickHenry! Unfortunately, such a statement of "mutual duty", if uttered today by a public official, would probably get him or her sued for violating "the separation of church and state." Or put him on the front page of the New York Times, and not in a flattering light (as in, "this person is an insensitive religious bigot"). That's how "rigid" many people are on this question. We seem to have lost our ability to draw reasonable distinctions about such matters.