David Theroux of the Independent Institute concludes his two-part article on “secular theocracy” here (the full article can be read here). In this second part, Theroux observes that “C.S. Lewis understood that natural law applies to all human behavior including government officials.”
Indeed, it is hard to see how the rule of law can function apart from a conception of the natural law. Now as Theroux shows, not just any conception of the natural law will do. It has to be one rooted in the divine lawgiver to those created in his image, with the implications for dignity and basic rights entailed by such.
Otherwise you might have a “natural law” that empowers the strong over the weak on the basis of their ability to dominate, or their intelligence, or their “fitness” to rule. See, for instance, Sam Gregg’s explanation of how Plato and Aristotle justified slavery.