It is the godless Constitution, which calls on no higher power than "We the People", that is the necessary and sufficient legal basis to form the social contract that was to create the American system of government. English Common Law had a lot of influnce also.
"As the government of the United States of America is not, in any sense, founded on the Christian Religion..." is a quote from the Treaty with Tripoli...words written by no less than George Washington in his second term and passed without fanfare during the first term of John Adams.
Many were deists. But all subscribed to a Judeo-Christian concept of morality, regardless of why.
When we say this was a Judeo-Christian nation, we don't mean that everyone was required to be a Christian or a Jew, we mean that everyone shared an understanding of what is right and what is wrong, and that this understanding came from the Judeo-Christian morality.
Shalom.