When the U.S. Constitution was forced on the American people in 1788 religion was not very different than it is today. There were no Mormons yet, but most of the major Christian sects had made an appearance. The educated ruling class studied the Greek and Roman classics, meaning they were exposed to both pagan beliefs and skepticism. Many Americans, possibly a majority were agnostics, atheists, materialists, or deists.
Politics was already a highly developed art and craft. So was its camp follower, corruption. I would argue that the first Congress under the new Constitution was the most corrupt in American history, but then there is a lot of competition, and records of corruption are incomplete, so I can't be sure. The U.S. Constitution was largely based on the British Constitution, but also borrowed from the Iroquois confederation, ancient Greek and Roman forms, and the various states, notably the State of Virginia.
Science, on the other hand, was just beginning its climb up Mount Sinai. Isaac Newton had invented calculus and outlined what is now called classical physics; he had died in 1727. But electricity was still something created by friction or lightning. Charles Darwin would not be born until 1809. Atoms would remain basically undifferentiated blobs until the 1890's.
For a document designed to centralize the power of the ruling elite, with just enough democracy included to keep the rabble from revolting too often, the U.S. Constitution is a pretty good basis for government. Its most obvious flaws, like the structure of the Senate, are a result of political compromise, not planning or wisdom. Amendments have, on the whole, improved it over the centuries. Most importantly, even the early ruling elite had their moments of working for the public good.
Largely because of Tea Party agitation, many Americans are taking a good look at their Constitution. Like any complex object, it can be looked at from many different angles. A citizen might like one part and dislike another. Throw in the often-mythologized stories about the (white) men who wrote the damned thing, and you have a lot of room for principled disagreements. Need more room, and you can just desert principles, which is a rather common (and often successful) political strategy.
Religious people, I mean the truly religious, project their religion into almost everything. It is almost a truism in certain circles that the U.S. Constitution was written by God. Now you and I know that is not true: God writes in Latin, not English. But the God-smitten won't have it any other way.
Which creates its own issues. Were Amendments to the Constitution written by God, or just the Original Constitution? Did God also write the Articles of Confederation? Why exactly did God give Congress the power to "promote the progress of science" [Article I, Section VIII, paragraph 8]?
You know why. He was preparing the way for his greatest prophets. Charles Darwin, James Clerk Maxwell, Max Planck, Albert Einstein, and Francis Crick, among others.