While this is a broad question, I think the answer to your question is hidden in the question itself.
People who are fine with relativistic ethics either on a personal or cultural level think it is impossible, unnecessary, or undesirable to justify an ethical system.
In other words, such views take your seemingly rhetorical question:
If a code of ethics is relative, then there is no absolute right and wrong and everything is only justifiable inside the appropriate point of reference.
And then they go for the other horn of the dilemma -- they accept that this means everything is only justifiable inside the appropriate point of reference. Generally, they then assert that this what has always been going on in ethics. Often this is done by taking either a sociological or biological explanation as the correct understanding of ethics.
As such, they accept that the only justifications you can usually get are internal (rather than external).
A philosopher (or maybe by the end of his career he would have preferred not to be called one) who accepted this view was Richard Rorty. At one point, he was president of the APA. He believes that when we imagine we're hitting bedrock with our thinking that we are just fooling ourselves.
More recently, Gil Harman and David Wong have both defended species of moral relativism.
Several other prominent philosophers disagree. A classic attempt to rebut this in the modern era is offered by James Rachel in his chapter entitled "Moral Relativism."
A rather generic argument follows the same course as your question but potentially amends it with questions about what moral relativism cannot condemn. Rachels for instance points out that the moral relativism cannot condemn slavery in either the past of his own culture or the present of any culture -- nor is it clear that he can condemn murders or holocausts in any of these.
Parts of your answer hint at a natural law approach to ethics. On such views, moral relativism fails because there is a natural moral order in the universe, and we are the sort of creatures endowed with the reason to see it.