Ironically one of the strongest influences the bible had on the constitution was the founders putting the "no religious tests are allowed as a qualification to hold any office" and "congress isn't allowed to make laws respecting an establishment of religion"
People really don’t understand how insane it was when each state had a state sponsored religion. Massachusetts once funded the Congregational Church, and only the Congregationalists, via tax dollars. We had Sunday business prohibitions until the 1990s.
So an influence, but less of a "these things are intertwined" sort of an influence and more of a "we don't ever want these two things to be mixed up with each other, we just fought a war so we can think whatever we want" sort of an influence.