Does the existence of moral diversity (i.e. widespread disagreement about moral issues, both between and within cultures) lend reason to think that there is no correct answer, or at least none that we are in a position to know?
We should begin by noting that superficial moral diversity is consistent with universal consensus regarding the fundamental moral principles. This is because the same base principles could yield very different results if applied in different circumstances or by people with different factual (non-moral) beliefs. This superficial form of moral "diversity" poses no skeptical threat. Such disagreements could be resolved by correcting factual mistakes and pointing out the situational differences. Only foundational disagreements have skeptical import, so the rest of this essay will concern this deeper form of moral diversity.
The mere existence of disagreement, even widespread and deep-rooted disagreement, is by itself insufficient to motivate skepticism. For note that the same phenomenon is found in the empirical sphere: regarding the origin of the human species, there is much disagreement both between and within cultures. But this doesn't lead us to suppose that there is "no truth of the matter", nor does it provide any reason to doubt the account provided by the theory of evolution. If others' disagreement is explicable through (say) ignorance or dogmatism, and poses no rational challenge to the evidential basis of our own beliefs, then mere diversity of opinion provides no skeptical impetus whatsoever. We should simply conclude that the others are mistaken.
Rational consideration of the empirical evidence will tend towards a scientific consensus. On those issues where even the experts are disagreed (e.g. how to reconcile general relativity with quantum mechanics), we at least think that progress is being made, and future discoveries will shed further light on the matter. It is less clear whether the same can be said of ethics. While we think that scientific disagreements cannot long survive the informed reflection of rational parties, it seems that moral disagreements more easily can. For example, significant and deep-rooted disagreements are found between consequentialists and deontologists, which cannot obviously be ascribed to ignorance or irrationality from either party.
If moral diversity could persist despite empirical omniscience and perfect rationality, then moral skepticism would be vindicated. Persisting diversity in such ideal circumstances would suggest that neither position was better justified than the other, or at least that the required justification is not accessible even to informed rational agents. We would have no reason to think that one position was more likely than the other to be true. We could not have moral knowledge -- our moral beliefs would be shown to be arbitrary and unjustified.
The reason behind this disparity between empirical and moral beliefs is presumably that only the former are responsive to reality. Empirical facts causally impact upon us, so our beliefs may respond accordingly. By contrast, whatever values are, they don't seem the sort of entity that could causally interact with our desires or moral beliefs. Positing some abstract Platonic realm of 'objective values' seems entirely superfluous to explaining the actual desires that we have. I think that this, and not moral diversity per se, is the real force behind moral skepticism.
But, as Michael Smith notes, our a priori beliefs are similarly disconnected from empirical reality. This leads many to think that the truth-maker for our a priori beliefs is merely constructive, i.e. that set of a priori beliefs that "we would all converge on if we were to subject our initial beliefs about what is a priori true to a reflective equilibrium process and so came up with a maximally informed and coherent and unified set of beliefs about what is a priori true."
Given the close analogy between evaluative and a priori beliefs, it seems plausible that their truthmakers rest upon a similar metaphysical foundation. That is, moral truths are merely constructive facts about what fully informed and rational agents would converge on, were they to subject their evaluative beliefs to a reflective equilibrium process.
On this view, the threat of moral diversity is not merely epistemological/skeptical, but metaphysical/nihilistic. If moral diversity would persist in ideal conditions, then not only would our moral beliefs lack justification, but there would be no moral truth at all!
Faced by this challenge, I think we have grounds for optimism. On a practical level, there is more moral consensus than is often realized. There are a vast number of simple ethical questions on which all reasonable people agree, from the general impermissibility of unprovoked violence, to the general desirability of compassion and promotion of human flourishing. This is often overlooked, as we tend to focus on the few most contentious moral issues (e.g. abortion), where judgments are often clouded by emotion or ideology -- or else just plain difficult. We tend not to notice the vast majority of moral life, which we navigate with relative ease.
Further, on a theoretical level, recent work has begun to bridge the gap between consequentialists and deontologists. The indirect utilitarians (e.g. Hare) have shown that there are good consequentialist reasons for adopting a deontological practical morality to guide our everyday moral thinking. And Derek Parfit has shown that the agent-relativity of strict deontological theories renders them collectively self-defeating. They must be revised at least part way towards the greater impartiality shown by consequentialism.
Ultimately, I think that moral diversity poses no greater skeptical challenge than the disagreements found in every other area of human discourse. As ethics is a field of great personal importance to our lives, it may be especially easy to be waylaid by emotion or wishful thinking. But if we are responsible in our epistemic practices, then the mere fact that others disagree with us need not mean that our moral beliefs are unjustified. Granted, it is possible that some moral disagreements really will prove to be ultimately irresoluble. But the often-overlooked commonplace moral consensus, and recent developments towards theoretical convergence, together provide some grounds for optimism.