At the New York Times, Appiah considers a similar question from a Jewish reader who does not want to be classified as white. Appiah notes that historically, Jews were often seen as non-white by other whites; they were seen as belonging to a different race altogether. Mein Kampf argues for this position.
Nevertheless, Appiah argues that (Ashkenazi?) Jews today qualify as white, because "white" is a social construction and the current social construction classifies them as such:
Let's consider two serious objections to this line of reasoning, one more disturbing than the other.
Smaller Objection, 1: It's not altogether clear whether current social practices treat white-looking Jews as white. Lots of KKK people and others regard them as non-white. Jews are still subject to discrimination and bias, and the people doing so often explicitly do so by thinking of Jews as a separate race. There are explicit anti-Semites in the UK Labour Party. BDS activists are often anti-Semitic. Etc. Again, some of this is motivated by regarding Jews as different race, not merely a different culture, ethnicity, or religious group.
So, empirically, it's more open than Appiah lets on just how white Jews are on current social constructions.
Bigger Objection, 2: Imagine objection 1 doesn't stand. Imagine instead that most everyone regards Jews as white, but this particular Jewish person, or perhaps many or most Jewish people, nevertheless reject the label. Appiah argues that since racial categories are social constructs, nevertheless, the prevailing social construct holds and so Jewish people are white.
However, this kind of reasoning has some uncomfortable implications. Consider that many argue that there is a distinction between sex and gender. Sex is a biological, scientific category. It is a natural kind. Gender, on the other hand, is a social construction. Thus, as many argue, gender is not determined by sex; a biological male could in principle be a woman, something other than a woman or man, or altogether genderless.
But Appiah is arguing that part of social constructivism is, well, social construction. Socially constructed categories are determined by, well, what society has constructed. So, he thinks, right now "white" is socially instructed to include (Ashkenazi) Jews, and thus they are white, even if they disagree.
But similarly, what "woman" means is a social construction. And so, on Appiah's reasoning, to determine whether someone is a woman, we need to see what the prevailing social construction is in any given society using that term.
Here's the worry, then. At least until very recently, the prevailing social construction in the West was that an adult biological female is a woman, regardless of what they think of themselves, how they dress, how they act, and how they perform their role. Same mutatis mutandis for "man". Appiah tells us that social constructivism means that when doing the metaphysics of race, we must defer to the actual constructions, and we have to check to see what they currently are. But then, presumably, by that same reasoning, at least until recently, every trans woman was actually still a man, and every trans man was still a woman. Even though logically-speaking, as philosophers pointed out long ago, gender does not equal sex, until every recently the two were nevertheless conflated in the actual prevailing social construction of gender.
I say until recently because from this sort of empiricist view of social construction, the social constructions are arguably changing. Right now, I'd say that there is no prevailing social construction in, say, the US. Among elites, there is one social construction; among the masses, a different one. What I mean by "woman" is not what, say, the typical Trumper means. Even this oversimplifies things.
In short, if Appiah's reader qualifies as white, then I am unsure why, say, Deirdre McCloskey qualifies as a woman overall, though she certainly does according to the prevailing social construction of gender within my elite sub-society. Perhaps in 20 years, when social constructions finish transitioning, she will qualify as a woman overall, but she did not by prevailing social constructions 30 years ago, and whether she does now overall is more puzzling.
Note that I am not endorsing this position; I am asking what Appiah's view/argument implies.
One might object that there is a difference. In one case--gender--it serious harms people to play along with the social construction, but in the other case--race--it does not. So in one case people have reason to rebel against the prevailing social constructions but not in the other.
This might be right--and indeed I have argued at length that people should not play along with socially constructed norms if the norms are bad and harmful. But this doesn't really resolve the metaphysical issue; it seems at most to suggest that a person in fact qualifies as one gender but should nevertheless feel free to deny this and insist on being treated as a member of another gender. Further, it's then an open question whether certain people--say certain Jews or Irish people--might also insist on not being called white (despite being so by prevailing constructions) on grounds that it harms them, offends them, dehumanizes them, erases their history, or whatnot.
Alternatively, perhaps Appiah thinks that gender is not socially constructed, but individually constructed. If so, why think that and why not say the same about race?