Deism isn't Christianity. Deism was a philosophical view that arose out of the European Enlightenment and, at least in part, was a reaction against organized religion. The Enligtenment championed reason over faith, and was a response to the wars of religion which had plagued Europe since the time of the Reformation. While not getting into everything on that; what is important to point out here is that Deism has nothing to do with Christianity.
I don't know that saying "Christian morals had a heavy influence on the Constitution" is as strong an argument as some would like it to be. I think people would like to think this is true, but isn't really.
I believe, at least historically, it far more accurate to say the Constitution is a product of the Enlightenement and the secular philosophies of the time. Such things can only be said to be "Christian" in the thinnest and most shallowest sense of the term, in much the same way that one can argue that the works of Shakespeare or Chaucer are "Christian" in that they arose out of a nominally Christian civilization. But to call nominally Christian civilization "Christian" is, itself, rather problematic and betrays the sanctity of our religion and the Lord whose precious blood has ransomed us from the powers and principalities of this fallen world.
-CryptoLutheran