No doubt the Framers were all deeply influenced by growing up in a land where virtually everyone is Christian. But you'd never know it from reading the Constitution. Other than the date at the end, there's no mention of religion, except to prevent religious tests for holding office. As we often point out in the science threads, the bible isn't a science book. Similarly, the Constitution isn't a theological work.
The Federalist Papers (mostly by Madison & Hamilton) are universally regarded as the most authoritative source for the intent of the Framers. You can search the Federalist Papers on line (I have done this), but you'll find not one mention of the words "bible," "scripture," or "Jesus." The word "Christian" appears once, in a reference to an historical period. "Lord" appears 5 times, but always in reference to aristocracy or the House of Lords. "God" appears 3 times, respectively refering to demi- gods, pagan gods, and nature's god.
Don't take my word for it. Here's a searchable copy: The Federalist Papers.
I agree. I don't think our Founding Fathers anticipated the cultural war we'd be fighting today, though. They'd have fought it another way, of that I'm certain. We need to start thinking not as they would, but as effectively as they would. This world is just as much on the edge as theirs was.