(Indeed, when Gentiles, who do not have the law, do by nature things required by the law, they are a law for themselves, even though they do not have the law. They show that the requirements of the law are written on their hearts, their consciences also bearing witness, and their thoughts sometimes accusing them and at other times even defending them.) NIV
The NIV bible has the better rendering of what Paul was arguing in this passage by enclosing it in parentheses rather than the ESV which conveys the idea of a natural law. The early Protestant theologians such as Calvin thought it was speaking about the Natural Law’s effect upon Pagan behavior. Like the NIV, I prefer to see the more contextual reading of Paul saying the behavior of Gentile Christians fulfills the Law, even though they do not follow all the minutia as a contemporary Jew who strictly follows the Mosaic Law.
Evidently there were Jewish “guides” (vss. 17,19) who wanted to instruct the believers, calling them “foolish” and “children” (vs. 20). They thought the Mosaic Law was “the embodiment of knowledge and truth” (vs. 20). Since the Gentiles were not versed in the Mosaic Code, they would guide these Gentiles the correct way. The Law was holy, righteous, and good but it couldn’t ultimately make the subject righteous; it only points out faults. The person was doing wrong themselves and looking down on everyone else (vss.21-23).
Paul is speaking generally that Christians are so holy and good without the formal Mosaic Law that, in the Judgement, it will be evident that they were written in The Book of Life (Rev. 20.12-15). These Gentiles had been adopted as children and were now acting out their renewed nature. This is what the term means (phusei): from themselves as to who they are in disposition or their constitution. Therefore, Paul is saying that God wrote the Law’s requirements on these Gentile Christian hearts, since He regenerated them. This is a parenthetical clause speaking in the immediate context of what the Judgement will entail (vss, 12-16).
Is there a Natural Law? I don’t know. This passage, however, is not speaking about that concept at all.