First, this isn’t necessarily an argument to completely refute the idea of moral relativism, but to raise doubts in it. And that is to point out that to hold onto moral relativism, one must accept that statements like “cruelty for its own sake is wrong,” “rape is wrong,” “genocide is wrong,” “compassion for others is a virtue,” and “parents have a duty to take care of their kids or put them in a position where they will be taken care of” are not true. In fact, moral relativism, to maintain consistency, must reject all these statements as these statements are asserting themselves to be universals because some other culture may, and in fact do, think things like rape and genocide and parents abandoning their children are moral goods. And I would argue this is not a desirable position to hold for anyone.
Second to address the common position that since societies have so many different moral positions, morality cannot be objective, I would say this conclusion does not follow from the premise. For example, you have the argument of whether or not God or gods exist. There is a plethora of differing opinions on the matter but that doesn’t mean there is no truth. Either there is/are God/gods or there isn’t, and either there is one God or many. These questions necessitate a true statement rather than a relativist response. Similarly, I would say this is true pertaining to morality. Simply because there’s disagreement on morality, does not mean the answer is that there are no moral truths. Perhaps the truths themselves are just difficult for cultures to ascertain through reason. Also, I would say there isn’t as much disagreement as one thinks, especially if we’re referring to moral values rather than the moral norms of society. For example, the philosopher James Rachels found that anthropologists discovered eskimos left unfit infants out in the snow. Now this is completely against the moral norms for most people in the west and east; however, the moral value behind this practice is shared by western and eastern societies. And that moral value is the preservation of human life. Given the harsh climate, taking care of every infant puts the family at risk of death and so they developed this practice. So while the moral practice is abhorrent, especially to us, they still share with us the same moral value. They simply manifested it differently given their differences in environment and rationale. Third, let’s presuppose that moral relativism is true. Say you have two individuals from different cultures interact with one another. One is a farmer who grows apples and the other is from a culture where they believe one cannot own the products of the earth. The one who believes you cannot own earthly products, believing it’s fair game due to cultural upbringing, takes an apple and eats it. The farmer sees this and they begin a dispute where the farmer is accusing the other of stealing and the other is saying one cannot own the products of the earth. Assuming moral relativism is true, both are correct. The farmer is correct in saying the other stole and the other is correct in that you cannot steal what is natural. But this agreement does nothing to absolve the issue and, to find a solution, some form of compromise (whether in favor of one of the individuals or split evenly) must be made. The same is true for every time there is a moral dispute and eventually you come to everything having a compromise. Essentially, even if relativism were true, societies would have to act as if it weren’t for cohesion and peace. Therefore, even if moral relativism were true, it would be completely impractical in the real world and only applicable in theory. And if it’s only applicable in theory, I question its validity. I have more I can say, but I’ll leave it here for now given the length.