America worships the Constitution and Bill of Rights like the Middle East worships the Quran.
I think the Constitution and the Bill of Rights have a different dynamic than religious texts; the real danger in America is that a lot of Americans worship a much-adapted version of the Bible.
The core belief of American government philosophy (at least my version of it; many Americans very clearly do not believe this) comes from the Declaration of Independence:
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. — That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, — That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness.
The main things here are (1) equality and (2) the notion that governments exist to serve the interests of the people, contra to the general notion of religious government philosophy that the role of government is to force people to behave however the religious doctrine says they should.