A Nation birthed by the Enlightenment - not Judeo-Christian, whatever that means exactly

Whenever I hear that the United States is a Judeo-Christian nation, I try to refer the speaker to our first documents. Read Federalist No. 10, read the writings of the founders, hell, read the Constitution. Nowhere is this stated. These documents conspicuously do not refer to any religion as their source.
In fact, here is a great quote dug up by Andrew Sullivan from The Conservative Soul:

In 1797, the U.S. Senate unanimously approved the "Treaty of Tripoli," an attempt to deal with Muslim piracy and terrorism in the Mediterranean. One of its clauses read:
"As the Government of the United States of America is not, in any sense, founded on the Christian religion; as it has in itself no character of enmity against the laws, religion, or tranquillity, of Mussulmen; and, as the said States never entered into any war, or act of hostility against any Mahometan nation, it is declared by the parties, that no pretext arising from religious opinions, shall ever produce an interruption of the harmony existing between the two countries."
It is hard to think of a leading contemporary Republican insisting that American government "is not, in any sense, founded on the Christian religion." In the early republic, not a single senator dissented.