If "natural" is the opposite of "rare" and means "common," it has no critical bite. Unless you think it's wrong to excel. If it's the opposite of "artifice" and means "what we haven't altered," it has no critical bite. Unless you think people shouldn't wear eyeglasses. If natural is the opposite of "supernatural" or "miraculous" and means "can be explained in the ordinary ways," it has no critical bite. Unless you think only divine intervention is wrong.
You could easily reverse the argument as well: if natural is "common" (throughout human history), then racial oppression and slavery are moral (indeed, that justification was present all over the place by equality opponents throughout the civil rights revolution). If natural means "the status quo," then it either means life is perfect by definition (because any alteration of the present is unnatural) or arbitrary (because we pick some random "past" status quo as what we should not have altered--presumably in that golden era before Roe and Engel but after Brown).
The problem, of course, is that the presence of something in reality has absolutely no bearing on its morality. Murder happens (is natural?), but that doesn't make it moral. In a democracy, for example, you will hear all the time that "X is right because 69% of the population believes it." But the presence of a democratic consensus has nothing to do with the morality of a given act--presumably people could agree to do immoral things. The end result of the reverse argument is a total abandonment of minority rights--if the majority says you should be dead, so be it. I don't think anybody conceptualizes morality like that.
Pseudo-Polymath points out that conservatives (it is mostly conservatives) who make this argument don't actually believe it, but are using it as a substitution for scriptural arguments which the presume will not persuade liberals (its interesting that they argue about the "naturality" of what they believe are supernaturally derived concepts--as Herzog points out, if natural means opposite of supernatural, then the only immoral being is God).
This is partially true, but misleading. The appeal to "naturality" is fundamentally past-based. That is, it draws on the experiences and guidance of prior events to direct how we should live our lives today. We've been doing X action forever, it is thus "natural" and should be continued. Nobody would argue that something recently developed is "natural," even when we agree it is a good thing (I have heard many justifications for racial equality, but never have I heard it called "natural"). Conservatism, as has been noted by many others, is about, well, conserving things. It does not like radical change, and takes comfort in the perceived predictability and knowability of the past. Hence, I do think that conservatives buy "naturality" as a valid defense of moral behavior beyond a mere transubstantiation of "the bible says so." PPM is still partially right, because many conservatives view the bible as the epitome of what is right and natural, so the arguments parallel. But presumably, one could be a Burkean conservative and argue on the basis of naturality without pegging it to biblical principles (though I'd agree that for most folks today, the Christian tail is wagging the Burkean dog).