While many people will admit that facts about the world do exist, it seems that many, especially highly educated people, will claim that ethical facts do not exist, at least no ethical facts that are true beyond a limited cultural context. Strangely, this often leads them to claim that the morality of other cultures should be respected and never judged on other terms than their own. This is obviously a self-contradiction since to claim that respect for the ethical codes of other cultures is always normative exactly is to claim that there is an ethical truth — that there is a claim about what we ought to do — that is true beyond a limited cultural context. So people who think of themselves as moral relativists actually tend to contradict the relativistic claim that no universal ethical truths exist, which reveals that they are, in fact, not truly moral relativists.
Unlike relativism with regard to all facts, moral relativism is not logically self-refuting per se, but assuming its validity does, however, prevent one from believing, and sincerely claiming, that we should act in one way rather than another — such a statement cannot be true according to a true moral relativist. And this includes statements about whether we should be moral realists or not, because even though moral relativism holds that moral realism is not true, the stance does not allow for the move that we therefore should not be moral realists. In making such a move, the purported moral relativist would stand revealed as a moral realist in disguise. Again, if one denies the existence of moral truths, one must, in order to be consistent, refrain from making purportedly valid claims about what we should do. This makes the moral relativist as valuable in a discussion about ethics as the relativist who denies that truths exist is in a discussion about any factual matter. The core statements that relativists make obviously keep them outside of discussions that concern the facts they claim do not exist. So while moral relativism is not logically self-contradicting, it does prevent one from saying that there is anything we should do. If a moral relativist does that that would be a self-contradiction.
Many self-professed moral relativists will insist that they can claim certain ethical statements to be true, but only within a given cultural frame. First of all, this is not moral relativism, but rather moral realism with a view of ethical facts one could call contextualistic in a hard sense. Secondly, such a position faces difficult problems of its own, especially the problem that faces any strongly contextualistic view of facts, namely: what demarcates one context, one cultural frame or setting, wherein one set of facts is true as opposed to another? A shared language? A shared history? A shared view of connectedness? This moral relativism, or, rather, hard contextual moral realism, rests upon a hard distinction between cultures that is as impossible to draw as it is to defend, especially in today's global, interconnected world.
So while moral relativism is not self-refuting per se, it does seem like it is hard, if not impossible, to consistently believe it to be true, since it seems humanly impossible, or at least impossible for any humane being, to really think that no actions are ethically right or wrong in any strong sense. As Sam Harris notes in his book, The Moral Landscape: “Given how deeply disposed we are to make universal moral claims, I think one can reasonably doubt whether any consistent moral relativist has ever existed.” 
Moral relativism is as harmful for our ethics as relativism with regard to all facts is for any science. It makes further progress impossible and rejects the progress we have made so far as truly being progress. And not only does moral relativism block the progress of the study of ethics — the study of what we should do — but also our moral practice itself, as it causes us to not see clearly on ethical matters. It prevents us from speaking out clearly against immoral actions, and it keeps us from raising ourselves to higher moral ground, at least from doing so as fast and intelligently as we easily could.
To the extent that we have made any progress in science, it has been because we admitted that there are facts to be uncovered and explained in the first place — because we admitted that relativism is false. Similarly, to the extent that we have made progress ethically, it has been because we have realized, in one way or the other, that some things truly are better than others. Admitting this is the only way in which we can intelligently strive toward ethical progress, because, otherwise, we cannot even talk about ethical progress — we cannot claim that the holocaust is really worse than a broken finger, or that we should avoid any of them, which a moral relativist cannot consistently say. As this reference to real-world examples exposes, moral relativism — the intellectual position that no state of the world is in any way better than any other; that happiness is not preferable over suffering and pain in any deep sense — is indeed nothing less than intellectually acquired psychopathy.
Unlike what the moral relativist erroneously thinks, we truly do have moral obligations, and one moral obligation that stands as clear as any is to go beyond moral relativism.
 Harris, S. (2010/2011). The Moral Landscape: How Science Can Determine Human Values. New York: Free Press. p. 45.
 It is also worth noting that those who have said the most interesting and useful things about ethics over the course of history, people like Aristotle, Kant, Bentham, Mill and Sidgwick, all happened to be moral realists in a strong sense, which is of course not surprising, as only moral realists consistently can say anything about what we should do. It is no coincidence that ethical wisdom has sprung from moral realism, while only the opposite has sprung from moral relativism.
 That moral relativism indeed does amount to intellectually acquired psychopathy is also captured well by the following quote from anthropologist Donald Symons that reveals the kind of absurdity that the attitude of apparent moral relativism commonly leads to:
If only one person in the world held down a terrified, struggling, screaming little girl, cut off her genitals with a septic blade, and sewed her back up, leaving only a tiny hole for urine and menstrual flow, the only question would be how severely that person should be punished, and whether the death penalty would be a sufficiently severe sanction. But when millions of people do this, instead of the enormity being magnified millions-fold, suddenly it becomes “culture,” and thereby magically becomes less, rather than more, horrible, and is even defended by some Western “moral thinkers,” including feminists.
Quoted from Pinker, S. (2002). The Blank Slate: The Modern Denial of Human Nature. New York: Penguin. p. 273.