The bible and the Constitution have nothing in common other than most of the founders believed in God and the bible.
Actually, many of the founding fathers were Deists and opposed the Bible. They believed in a creator, but that he did not concern himself with the daily lives of humans, and does not directly communicate with humans, either by revelation or by sacred books.
"I have examined all the known superstitions of the World, and I do not find in our particular superstition of Christianity one redeeming feature. They are all alike, founded on fables and mythology."
Thomas Jefferson
"I do not believe in the creed professed by the Jewish Church, by the Roman Church, by the Greek Church, by the Turkish Church, by the Protestant Church, nor by any church that I know of. My own mind is my own church."
Thomas Paine
"The doctrine of the divinity of Jesus is made a convenient cover for absurdity."
John Adams