One difficulty with relativism as popularly interpreted is that its proponents, its talkers, do not appear to be walking the walk; that is, the notion that there are no universals is itself proposed as a universal rule. This is an immediate and obvious contradiction. Equally obviously the proposal that all such rules are universals is logically consistent but does not appear to map very well to the real world.
We are left with the likelihood that some such rules are universals and some are not with no real way to determine which is which within the logical framework of the system. This is a problem as old as analytical philosophy, not just within the sub-field of ethics.
One can attempt to build a fairly relativistic ethical system with more or less coherence. Jeremy Bentham proposed that an action is to be measured by its ability to provide the most good for the most individuals affected. A little consideration will reveal that this refers strictly to a choice between alternative actions. One treats the analysis of a single action by measuring its effects against the consequence of inaction, still a relativistic approach.
But these are purely theoretical considerations. Clearly one undertake this moral calculus for every action taken; one would be paralyzed into an inevitably unethical inaction. In practice, one uses mores for this. My point is that even relativistic theoretical philosophers do so despite their insistence that it is on a fundamental level illegitimate. Even complete nihilists do this. It isn't how they think that they think, but it is how they behave. Even nihilists and relativistic philosophers obey such arbitrary conventions as traffic lights. If they don't, they die.
It is popularly thought that human behavior is to be corrected by theory, but I cannot see that it is not equally necessary to modify theory by considerations of actual human behavior. People may be behaving the way they do for considerations outside the confines of a theoretical model. Edmund Burke touched on this with respect to political philosophy. Any ethical system that is incapable of accounting for this sort of ad hoc behavior is inherently incomplete.
And, in practice, citing God as an authority for ethical mores may seem arbitrary but is no more arbitrary than citing theory. Equally arbitrary is the assertion that there are no legitimate authorities at all. "Arbitrary" therefore is a bit of a red herring - you can't escape it, the only difference is how you deal with it.
Oopsie - “one undertake” should be “one cannot undertake.” Poor proofreading.