The defenders of natural law cite Rom 2:15, as Dr. Beckwith did, saying that the law is written on our hearts. But I think they are missing something. Here is Rom 2:14 and 15:
14 For when Gentiles, who do not have the law, by nature do what the law requires, they are a law to themselves, even though they do not have the law. 15 They show that the work of the law is written on their hearts, while their conscience also bears witness, and their conflicting thoughts accuse or even excuse them
I may be splitting hairs here, but when I read those two verses I do not see that the specific content of the law is written on people's hearts, which would be required if natural law advocates are correct. In fact is specifically says that the Gentiles "do not have the law."
Instead, what I see is that the idea of law is written on people's hearts. What that means is that everyone has a sense of right and wrong, that some things are good and some things are evil. That does not say that everyone will agree with the specific content, or forms that sense may take. C.S. Lewis has an excellent analysis of this in the chapter entitled, "Right and Wrong as a Clue to the Meaning of the Universe," in his book, Mere Christianity.
Where Beckwith says that "the human mind is ordered toward the truth..." I would say merely that the human mind is ordered toward truth. Again, that seems to be nit-picking, but the insertion of "the" suggests that there is some positive (in the legal sense), knowable content. By removing "the" I am arguing that while all humans know that "truth" exists, no content is available to us through reason alone.
What natural law advocates seem to want to do is—as Beckwith accuses Hart of doing by stealth—to "derive an ought from an is (i.e.. nature)" to assist in establishing principles of law that all would have to agree were self-evident in nature. But nature does not cooperate with us in that task. Much of what we perceive as nature is a result of the Fall. But nature, as Tennyson famously reminded us, is "red in tooth and claw." (I accept that is a far too simplistic understanding of Beckwith's position.)
Even if nature were not corrupted by the Fall, our own reasoning is corrupted, and fundamental to the human condition is the idea that we are incapable of rightly perceiving good and evil without revelation—and in this case, I don't believe general revelation is sufficient, hence we have the Scriptures.