It is often pressed by evangelicals, and religious “scholars,” and local school boards [generally in the South], that the Constitution and the foundation of the American Republic come straight from the Ten Commandments, the Decalogue of Exodus. Since I regularly re-read Madison's Notes of Debates in the Federal Convention… and the Federalist Papers, and have more than a passing knowledge of the Bible, I suspect I am as good an authority on that claim as anyone, so let's examine the theory.

1. “I am the LORD your God, who brought you out of the land of Egypt, out of the house of bondage. You shall have no other gods before Me.” Well, there is no discussion of that in any of the founding documents, so let's move on.
2. “You shall not make for yourself a carved image—any likeness of anything that is in heaven above, or that is in the earth beneath, or that is in the water under the earth; you shall not bow down to them nor serve them. For I, the LORD your God, am a jealous God, visiting the iniquity of the fathers upon the children to the third and fourth generations of those who hate Me, but showing mercy to thousands, to those who love Me and keep My commandments.” Uh…, neither the Founders nor the Framers said much of that in erecting the Republic nor debating the Constitution, so let's move on.
3. “You shall not take the name of the LORD your God in vain, for the LORD will not hold him guiltless who takes His name in vain.” Hm…, they didn't really debate that point. It's interesting to note that taking the Lord's name “in vain” is quite different from cussing, about which we need not expand here, but the Framers didn't debate either.
4. “Remember the Sabbath day, to keep it holy. Six days you shall labor and do all your work, but the seventh day is the Sabbath of the LORD your God. In it you shall do no work: you, nor your son, nor your daughter, nor your male servant, nor your female servant, nor your cattle, nor your stranger who is within your gates. For in six days the LORD made the heavens and the earth, the sea, and all that is in them, and rested the seventh day. Therefore the LORD blessed the Sabbath day and hallowed it.” The Sabbath, of course, is Saturday; the debates raged on on Saturday; the Framers took Sunday off, but there was no discussion that the “Sabbath” clause had anything to do with the Republic and its foundations.
5. “Honor your father and your mother, that your days may be long upon the land which the LORD your God is giving you.” That is a marvelous and appropriate standard for civilized society of all government forms worldwide, but there was no discussion that it was a foundation for the Republic.
6. “You shall not murder.” That's a common law and Roman law standard greatly preceding the Republic and the Revolution and it was nowhere discussed as a foundation for the Republic, because it is a universal.
7. “You shall not commit adultery.” Calling Presidents Franklin Roosevelt, Eisenhower, Kennedy, Clinton, Trump, Reverend Jimmy Swaggart: your thoughts? That too is a common law and Roman law and religious law standard greatly preceding the Republic and the Revolution and it was nowhere discussed as a foundation for the Republic, because it is a universal, although “respected” only in the breach by many chief executives, inter multa alia.
8. “You shall not steal.” That too is a common law and Roman law standard greatly preceding the Republic and the Revolution and it was nowhere discussed as a foundation for the Republic, because it is a universal. Of course, violating it dovetails #10 because to expand we coveted others' land and then stole it. And we did that partially by bearing false witness against its owners [#9]. Indeed, violating #s 8, 9, and 10 is how Hawaii became American territory!
9. “You shall not bear false witness against your neighbor.” Yeah, it would be nice if government leaders would not lie about others, but they seem incapable of adhering to that universal ideal regardless of political leanings, but it was nowhere discussed by the Founders or the Framers in setting up the Republic.
10. “You shall not covet your neighbor's house; you shall not covet your neighbor's wife, nor his male servant, nor his female servant, nor his ox, nor his donkey, nor anything that is your neighbor's.” Goodness, coveting what others have is the very cornerstone of the establishment of the Republic and of its westward expansion, which began with the geographically challenged Christopher Columbus [whose creative ineptitude we celebrate with a national holiday], and to perfect that covetousness we bear and have borne much false witness against many neighbors. This certainly is not a foundation for the Republic, however often its breach is.

No, the celebrated Decalogue did not enter into the foundation of this Republic, although a couple of its standards are congruent with standards in play both then and now. Whether posting them in public schools violates the First Amendment is another question not discussed here, but asserting that they undergird the Constitution is constitutional sacrilege pressed only by those ignorant of the founding premises and debates about the Republic.