R. J. Snell, writing for Public Discourse, tries to answer the question of whether natural law is persuasive to anyone not already convinced:
First, natural lawyers needn’t convince or persuade anyone, for in an important way natural law cannot be proven—law is the condition of intelligible action. Instead, our task is to have our interlocutors pay attention, not to us and our arguments, but to themselves. My failure to persuade someone of the natural law happens only if they (a) will not pay attention to themselves, or (b) if they do not understand themselves. Of course, I have neither power nor responsibility over another’s capacity to know themselves, so I’ll politely decline any such obligations for my ethics. In fact, since natural law is pre-theoretical, depending not on a system of concepts but rather on self-understanding, the more I grant that a theory must be able to persuade, the more that alternative theories such as deontology or utility fall while natural law survives, for they actually must provide a proof whereas I make no such claim, asking only for my interlocutor to advert to his performance.
But those who aren’t yet convinced about natural law do treat it as just another theory among others. So it seems that anyone who isn’t already on board with natural law will probably not be convinced, especially if they’re looking for persuasive reasoning.