No Constitution because the word "God" isn't mentioned, but I think documents from Jefferson, Lincoln, and our nation's anthem is pretty strong (even if they were written centuries ago) proof that God is a big part of our country (and clearly our beginnings).
Diests still belived in God, don't forget, -FMA. Maybe not a Christian god, or a Catholic god, or an Islamic god, but they believed in a greater being, who they called "God" for sake of a better word.
They weren't anti-religion or anti-God, which is what you're trying to lead us to believe.