As Michael Smith writes in The Moral Problem, p.159:
Suppose we take a whole host of desires we have for specific and general things; desires which are not in fact derived from any desire that we have for something more general. We can ask ourselves whether we wouldn't get a more systematically justifiable set of desires by adding to this whole host of specific and general desires another general desire, or a more general desire still, a desire that, in turn, justifies and explains the more specific desires that we have. And the answer might be that we would.
It might help to offer an example. Imagine someone who thought three foods all tasted equally good, but he only desired to eat the first two. This desire set makes little sense. We require some explanation for why the third food is different, so as to justify its different treatment. If no such explanation is forthcoming, then the desire set is open to rational criticism for making arbitrary distinctions. It would make more sense for the agent to desire all the good-tasting foods (again, assuming there was no hidden reason for treating some differently). [M. Smith, 'In Defence of The Moral Problem' in Ethics and the a priori, p.269.]
Moreover, our desires ought to cohere not just with each other, but also with our evaluative beliefs. Suppose we believe that we have normative reason to Φ (that is, on Smith's analysis of normative reasons, "we believe that we would desire to Φ if we were fully rational") and yet we fail to desire to Φ. Then we are irrational "by our own lights. For we fail to have a desire that we believe it is rational for us to have." (TMP, p.177)
Now for the crucial question: does the amoralist's desire set suffer from incoherence? We've already established that he rationally ought not be purely egoistic, i.e. he must see at least some other people as having intrinsic worth as ends in themselves. But why should he desire the well-being of some people but not others? Unless he can draw some principled distinction between the groups, then the amoralist's desires are unacceptably arbitrary. The moralist's universal concern shows greater coherence and unity, and thus is less open to rational criticism.
I should note that this ties in nicely with Nagel's dissociative argument for agent-neutral reasons. We all recognize that pain is bad. But by denying agent-neutral reasons, the amoralist is committed to the view that no-one else actually has any reason to relieve his pain. This seems implausible, assuming we have reason to do anything at all.
Once we recognize that our own interests matter, and that other people are relevantly similar to ourselves, then consistency would seem to require us to conclude that the interests of other people matter too. As Christine Korsgaard notes, when we are asked to put ourselves in the position of a victim of our cruelty, we do not respond, "Someone doing that to me, why that would be terrible! But then I am me, after all." ['The Sources of Normativity' in Darwall et al. (eds.) Moral Discourse and Practice, p.400] We all recognize that something is no less terrible merely because it happens to someone else.
So we are now in a position to respond to the amoralist. It seems that someone could fail to care about others without making any empirical or logical mistake. But they show a degree of arbitrariness and incoherence in their values which is open to rational criticism. The amoralist must retreat to a form of rational nihilism, denying that there are any agent-neutral reasons for action. ("My pain isn't really bad. It's just bad to me.") But this position squares poorly with our intuitive judgments about the badness of pain and the impersonal reasons that it gives rise to. If we're right to hold that there can be agent-neutral reasons, then it follows that the amoralist is irrational in failing to recognize and act on these reasons. But even if we grant reason-relativism, we can still fault the amoralist for drawing arbitrary distinctions by caring about some people but not others who are relevantly similar. They could make their desire set more coherent by expanding their circle of concern to cover everyone (perhaps gradually 'fading out' as more 'distant' people would be slightly different in ways that justify having slightly less concern for them). So, either way, the amoralist is open to rational criticism.