tl;dr - "appeal to nature fallacy" does not have the magical power you suggest. Whether there's a problem depends on who decides when looking to nature is fallacious which is usually inspired by other commitments.
There's two cross-considerations here before we can decide whether a natural law approach to rights is fallacious. First, we need to understand what it means to call something a "fallacy." Second, we need to take what we gain in that discussion and see whether deriving rights from observing nature is (a) absolutely a commission of the naturalistic fallacy, (b) potentially a commission of the naturalistic fallacy, or (c) absolutely not a commission of a naturalistic fallacy.
There's two basic types of fallacies. There's fallacies in formal logic. These are pretty cut and dry within formal logic. For instance, "affirming the consequent" is just plain formally invalid. Similarly, the pattern A or B. A. Therefore B is fallacious. Fallacies in the real of informal reasoning are less cut and dry in both their definition and scope. The formally valid hypothetical syllogism ("chain rule") is also roughly the same as the informal fallacy "slippery slope." Whether something is a valid application of the chain rule or an invalid slippery slope is going to be a matter of judgment and application rather than something that everyone sees and agrees on. (For that matter, if an argument is offered informally rather than formally, some of the formal fallacies will be harder to stick to it to).
The fallacy "appeal to nature" is by all accounts an informal fallacy. This creates two problems. First, while often used, it's not perfectly clear what the fallacy means. On some level, we all agree that it's the illicit appeal to nature to resolve a dispute that doesn't belong to nature, but the question is what makes an appeal licit or illicit.
To give an example, if we want to know how far a bullet travels in 1 second, there's two ways to figure it out: (1) calculate it using equations and (2) fire the bullet and measure the distance it traveled in one second. If (1) and (2) disagree, it's an appeal to nature (physics in this case) to get our answer from (2), but no one would say this is fallacious.
In general, people view appeal to nature as fallacious when we take nature as a (naked?) guide for human conduct. For instance, the role of nature is a major component in debates about (a) marriage in general, (b) sex acts,(c) whether people should have children, (d) infanticide, etc.
The general pattern is something like this:
- X happens in nature
- Therefore, X should be a rule for human conduct
But that doesn't mean everything that involves premise 1 and conclusion 2 is going to be something we all agree is fallacious. For instance,
- Animals need nutrition.
- Animals care for their young (especially infants, pups, etc. that cannot care for themselves).
- Humans are animals.
- Therefore, it is cruel for human parents to not take care of their infants.
Is this the "appeal to nature" fallacy? I don't really think so. Whereas I'd probably call this an "appeal to nature" fallacy:
- Panda mothers usually only care for one twin.
- Therefore, human mothers should only care for one twin.
Why would I consider the latter fallacious but not the former? Well in large part, because I take the former pattern to be something we see occurring in natural that would be good to copy for a variety of reasons including the sort of animals we are. In the latter, I see an inept attempt to hamstring nature into a source of morality.
Could I give a more formal explanation? Sure, but the point here is two-fold: (1) there are some clear cases where nature is a source -- such as in physics. It does not matter what our equation says if the results say otherwise. And in such cases, not looking to nature is fallacious. (2) there are people who'd call both fallacious appeals to nature. As an informal fallacy, the origin may lie elsewhere. For instance, if they are committed to the claim that nature has no relation to morality. In other words this is an open area of dispute but the real complaint "hey, he looked at nature to get a source of value" is a complaint that only one side finds convincing.
In the case of human morality, many major philosophers are committed to seeing nature as one source for how humans should live. This is a key feature of Aristotle's political and ethical thought. Specifically, he thinks individual humans are parts of a larger polis, and to some extent we are to be guided by the fact we are animals that form these sorts of city-units or rather we are city-units made up of individual animals. Aristotle does not however talk of rights.
Rights talk is all basically post Magna Carta. But the idea for the earlier rights theorists was that when we look at the pattern of nature, we can with the application of good perception and reason identify certain features of ourselves that imply rights. (Or for some, though perhaps less palatable now, we can probably directly perceive indelible rights through being created).
Hegel has a modified version of Aristotle's vision of society. In Hegel's case, nature does not give us rights, but nature does give us some of the data we rationally process and turn into rights.