The late Justice Scalia was one of these legal positivists. He was a devout Roman Catholic who personally believed in what his Church taught: the natural law of the Aristotlean-Thomistic tradition. His point was if legislatures wish to use that to inform their conscience when drafting and voting for legislation they are free to do so. Just as they are free to use the the Bible, Book of Mormon, the works of Immanuel Kant or whatever they wish.
I can't resolve the debate between the legal positivists and those who believe that a higher organic law undergirds our system and can be used in constitutional interpretation. Legal positivism predominates. I do believe that the reason why some very distinguished scholars dissent and believe in natural law and rights is that quotations abound from America's Founders demonstrating that they believed in the existence of such.
Christian nationalists think that it's Christianity that is the source of this higher law. After years of studying this, I don't think that's right. Though God does have a place in documents like the Declaration of Independence and Alexander Hamilton's "The Farmer Refuted," which are among the sources of the aforementioned quotations by America's Founders on their belief in the existence of the "brooding omnipresence in the sky."
We can add William Livingston's name to the list of Founders who denied that Christianity was the source of the omnipresence. Livingston was a framer of the US Constitution and played a key role in getting the document ratified in New Jersey.
The denial of Christianity as the organic law of America comes from Livingston as he commented on the Articles of Confederation. The Articles do invoke "the Great Governor of the World," but that wasn't good enough for John Mason who wanted language that the "law of the eternal God, as contained in the sacred Scriptures, of the Old and New Testament, [is] the supreme law of the United States,..."
That language is also absent from the US Constitution. In fact the US Constitution, unlike the Articles, is Godless. Below is from Livingston's letter to John Mason, Princeton, 29th May, 1778:
And to have made the 'law of the eternal God, as contained in the sacred Scriptures, of the Old and New Testament, the supreme law of the United States,' would, I conceive, have laid the foundation of endless altercation and dispute, as the very first question that would have arisen upon that article would be, whether we were bound by the ceremonial as well as the moral law, delivered by Moses to the people of Israel. Should we confine ourselves to the law of God, as contained in the Scriptures of the New Testament (which is undoubtedly obligatory upon all Christians), there would still have been endless disputes about the construction of the of these laws. Shall the meaning be ascertained by every individual for himself, or by public authority? If the first, all human laws respecting the subject are merely nugatory; if the latter, government must assume the detestable power of Henry the Eighth, and enforce their own interpretations with pains and penalties.Post Script: For the full, easily accessible version of Livingston's letter, see here.
[A]nd the inseparable connexion between the morals of the people and the good of society will compel them to pay due attention to external regularity and decorum; but true piety again has never been agreed upon by mankind, and I should not be willing that any human tribunal should settle its definition for me.