As for the Constitution. It never, ever mentions God, the Creator or any such thing. I've posted this several times. See for yourself.
http://www.archives.gov/exhibits/charters/constitution_transcript.html
The only place in the Constitution that religion is even mentioned is in the Bill of Rights. It is mentioned with freedom of speech, and press.
http://www.archives.gov/exhibits/charters/bill_of_rights_transcript.html
Also, virtually every religion has provisions about morals, and most are fairly similar at their cores. Treat people decently, don't lie, cheat, steal, or kill, worship god,etc.
Freedom of worship also means the freedom from worship. In the case of laws, this means we are not subjected to one religion's version of the truth. This means any argument for or against something that starts with "God/The Bible/The Koran/etc says" is not legitimate legal argument.
The Constitution of the US says I don't have to follow religious laws, and quite frankly I wouldn't want to live in a theocracy.