This tradition sees morality as a matter of the moral laws that follow from what fundamentally makes us human: our human nature.
This is extraordinarily misleading. The natural law tradition sees at least some morality as a matter of the moral laws that are rationally evident in themselves, or that follow from such laws. Natural law theorists have historically held that there are areas of moral life only loosely related to moral law; it's just that these looser areas presuppose the moral laws as a skeletal framework. Natural law theory is also not itself a theory of human nature; it is a theory of practical human reason where it concerns the common good of human beings. Human nature enters into the picture because the kinds of goods that can be common goods for all human beings obviously have to be identified in light of human nature itself. Gutting then goes on to suggest:
The problem is that, rightly developed, natural-law thinking seems to support rather than reject the morality of homosexual behavior. Consider this line of thought from John Corvino, a philosopher at Wayne State University: “A gay relationship, like a straight relationship, can be a significant avenue of meaning, growth, and fulfillment. It can realize a variety of genuine human goods; it can bear good fruit. . . . [For both straight and gay couples,] sex is a powerful and unique way of building, celebrating, and replenishing intimacy.” The sort of relationship Corvino describes seems clearly one that would contribute to a couple’s fulfillment as human beings — whether the sex involved is hetero- or homosexual. Isn’t this just what it should mean to live in accord with human nature?
But natural law theory is not specifically about "fulfillment as human beings", a vague phrase that describes almost every moral account one could name. Corvino's argument is entirely focused on consequences, but no natural law theorist would look only at consequences. And, what is more, the goods identified by Corvino's argument are all arguably private goods: meaning, growth, fulfillment, intimacy -- while these might touch on or be relevant to goods common to all humanity, nothing about them requires that they be common goods in themselves. Contrast this with reproduction as continuance of the human race or reason itself, which are reasonable candidates for a good that is common to all human beings. Your private meaning, growth, fulfillment, and intimacy only become even plausibly significant for determining right and wrong in a genuine natural law theory to the extent that they become entangled with something like reproduction or the goods of reason itself. Now, in a matter like sex, as with politics and religion, this can happen fairly easily, since sex is a major pillar of civilized life; but it has to happen.
Gutting then attributes a supplementary argument made by some natural law theorists (the selfishness point he mentions) to all natural law theorists as if it were their primary argument, and manages to mangle it, as well, focusing on selfishness rather than objectification. In the course of this mangling, he says:
The awkward talk of “an act that could not in principle result in pregnancy” is necessary since those who put forward this argument want to maintain that heterosexual unions in which one (or both) of the partners is sterile are still moral. There’s nothing unnatural about their intercourse because it’s the sort of act that in general can lead to reproduction.
Here we see Gutting falling into a common misapprehension, that natural law theorists arguing against same-sex sexual activities do so because they are claiming that the intercourse is "unnatural" (unless by this you simply mean 'not appropriate to a rational life'). Natural law theorists rather argue that they are wrong on the same ground that they would argue that it is wrong to quarrel with people just because you feel like it: that it is in a robust sense irrational in the sense established by the account of practical reason in natural law theory. They may sometimes also hold (and often have held) that certain problematic features of the wrong are aggravated because it is unnatural in some specific sense; but this is a distinct issue, and the specific sense matters.
All of these obscurities and confusions are rookie mistakes; they are not what one would reasonably expect of a professor of philosophy at Notre Dame. Despite the reference to Corvino, for instance, Gutting's argument on the subject is massively more muddled than the arguments of Corvino himself, even when Corvino is also talking in a very limited and popular forum. Contrast Gutting's argument, for instance, with Corvino's argument in this seven-minute YouTube comment; despite the fact that they are very similar in many of their basic ideas, Corvino's argument in the last three minutes alone is massively better than Gutting's: less vague, more coherent, better reasoned.
[I found, incidentally, the 'many would see the argument as proving too much' line a bit amusing, since this is how one plays rhetorical games: if it were narrowly focused, the argument would be dismissed as biased against gays and lesbians; since it covers "birth control, masturbation and even non-reproductive sexual acts between heterosexuals" (you have to love that "even", as if sexual acts between heterosexuals were somehow especially immune to ethical scrutiny), and, indeed, only addresses homosexual sexual activities as a secondary and incidental matter, one instead casts the net as widely as possible to get as many allies as possible. In reality, of course, any Catholic account of sexual ethics is going to be fairly restrictive across the board because (1) it will maintain the central importance of marriage between man and woman; and (2) it will preserve essential elements of the ethical ideas of Classical antiquity, which also tend to be quite restrictive in sexual matters (since even when they tolerate things that later ceased to be tolerated, the philosophers of antiquity were constant critics of common indulgences of their culture). Expecting a Catholic position on sex that does not argue for what "many would see" as "proving too much" would be absurd; and it is irrelevant to the question of what a Catholic position can accommodate, which is the question at hand.]