Jeff Ely takes issue with this. The Mankw-Weinzerl result, he says, does not depend on a utilitarian welfare function, and so is more general.
It has nothing to do with utilitarianism. (And your natural objection to taxing height therefore says nothing about your attitudes toward income redistribution.)
I think Ely is right, but only partly so. Yes, it is not utilitarianism per se that justifies the height tax; it is consequentialism (the idea that only outcomes that matter for assessing outcomes, not how those outcomes come about), of which utilitarianism is but one example. But if you do reject the height tax, you are rejecting a welfare framework that assesses all policy purely through the eyes of an equity-efficiency trade-off. There is still room for income redistribution in an expanded framework that includes non-consequentialist values, but that framework needs to be articulated. I read Mankiw not as saying that rejection of the height tax implies rejection of any redistribution, but rather as saying that economic policy analysis needs to get beyond Bergson-Samuelson welfare functions and think about alternative frameworks. While I don't necessarily agree with Mankiw’s just-deserts alternative, it does go someway to accepting that challenge.