Without a concept of God then, our Constitution and Bill of Rights are both meaningless.
Why's that?
And, the second part of your supposition is a non-sequiter. The Constitution makes no mention of God and is actually patterned on the old Roman Republic (read: pagan) concept of a government of laws. Remember, the Founding Fathers were, for the most part, classically trained. The Constitution also owes quite a bit to 17th- and 18th-century natural philosophy. The latter holds that rights are inherent within an individual because holding that rights are bestowed by God means they are gifts, and not rights, and may be taken away by the Almighty or anyone supposedly representing the latter.