It's a good point and worth exploring because such appeals to nature are made in numerous debates. Pope Benedict for example used his Christmas speech to attack theories of socially constructed gender :
People dispute the idea that they have a nature, given by their bodily identity, that serves as a defining element of the human being. They deny their nature and decide that it is not something previously given to them, but that they make it for themselves. According to the biblical creation account, being created by God as male and female pertains to the essence of the human creature. This duality is an essential aspect of what being human is all about, as ordained by God. This very duality as something previously given is what is now disputed. The words of the creation account: “male and female he created them” (Gen 1:27) no longer apply. No, what applies now is this: it was not God who created them male and female – hitherto society did this, now we decide for ourselves. Man and woman as created realities, as the nature of the human being, no longer exist. Man calls his nature into question. From now on he is merely spirit and will. The manipulation of nature, which we deplore today where our environment is concerned, now becomes man’s fundamental choice where he himself is concerned. From now on there is only the abstract human being, who chooses for himself what his nature is to be. [link]
In general, religious people of various sects and traditions like to claim their dogma is in symphony with nature, reason and the commandments of various Gods. As Neiman notes, it's an old move but how do we determine what's truly natural considering the sheer diversity of the world and our current 'denatured' state of living? Does the term 'denatured' even make sense? Aren't human cities as natural as a bees hive or a badgers den? How do we distinguish what is truly human nature and what is mere social convention?
What is human nature? In most cases people simply project their own views into the question then scurry around looking for suitable evidence. Is human nature inherently good or inherently twisted (say through original sin)? Universal or crucially different depending on race?
During the Enlightenment the practice of conveniently plucking evidence from other cultures to support your own view point was so prevalent that the Marquis de Sade parodied the practice : when he wanted to defend xenophobia, he invented African tribes so hostile to invaders that they rejected their own corpses!
Be weary of appeals to human nature: We tend to become hopelessly partisan by projecting our values into what we want to see, an annoying habit leading Rousseau to recommend: "Let us begin by setting aside all the facts, for they do not affect the question". Radical honesty was Rousseau's recommendation: Given that we cannot currently reliably separate human nature from social conditioning, instead of wrapping the evidence to fit our preconceived opinions and claiming certainty, why not instead test for plausibility and admit the account is designed to further the moral and political viewpoint you are defending?
Pope Benedict is claiming knowledge he simply cannot have. We do not 'deny' we have a nature, but the picture is more complicated either Roman Catholics or feminist 'blank slate' theorists are comfortable admitting.