Rehnquist and Scalia, Rehnquist more than Scalia, have been cautious about bringing natural law into their decisions, whatever they may privately believe (or have believed). Clarence Thomas seems to be the only justice who presently inclines in that direction.
But Scalia does appear to recognize in this article that if you believe that "law" is just "there," to be found and interpreted, then either it's in some vague nowhere in the empty aether or it comes from God. Earlier generations found it reasonable to say that it came from God, or at least was up in the sky somewhere in some Platonic realm of ideas.
Once you abandon that conviction, then there is nothing but the text to work with. But it's difficult to work with just a text. So people turn to supporting background materials. What did the Founders privately intend? Scalia argues that is irrelevant. What did the words mean to an educated, average person at the time? Scalia argues that is the best approach.
Proceeding from that assumption, it seems to me that "Nature and Nature's God" must inevitably enter into the way you read the text. If you leave that out, then you will inevitably come to read the words in a modernist and then a revisionist spirit.