The Founders and the Framers Adamantly Did Not Intend to Make the Ten Commandments Law
Thomas Jefferson specifically railed against attempts to claim that the common law incorporated the Ten Commandments when he criticized judges for "lay[ing] the yoke of their own opinions on the necks of others by declaring that [the Ten Commandments] make a part of the law of the land." John Adams also questioned the influence of the Commandments and the Sermon on the Mount on the legal system.
At the Constitutional Convention, the Framers looked to the examples of antiquity--the Greeks and the Romans - and
not to the Ten Commandments. They were a pragmatic lot, and they were not interested in being bound by their religious heritage (despite today's claims to the contrary)
Those Framers who were well educated had studied antiquity and the classics in depth (unlike the vast majority of Americans today, even those who are college educated). Thus, they were perfectly comfortable borrowing and adapting whatever suited their purposes. It would be a huge overstatement to say that they felt themselves constrained by the four corners of the Bible in finding the right government, or setting up the ultimate law that would rule the U.S.
The sources that influenced the Framers ranged from Greek and Roman law, to John Locke, to Scottish Common Sense philosophers, to Grotius. The influence of the Common Sensists was quite evident in the Framers' strong belief in the power of reason--not revelation or Biblical passages--to determine government. They were also influenced by the dominant religion of the time--Calvinism--in the sense that their world view was rooted in distrust of any human who holds power. And this list is only a beginning.
Meanwhile, the very tenor of the times was distrustful of organized religion, and especially stakeholder claims to truth by religious individuals. Madison declared, in his Memorial and Remonstrance of 1785, "experience witnesseth that ecclesiastical establishments, instead of maintaining purity and efficacy of Religion, have had a contrary operation. During almost fifteen centuries, the legal establishment of Christianity has been on trial. What have been its fruits? More or less in all places, pride and indolence in the Clergy, ignorance and servility in the laity, in both, superstition, bigotry and persecution."