Here's a thought experiment that I sometimes give to my students if I'm teaching ethics.
Suppose that you have a fundamental moral disagreement with someone very close to you, say a wife or husband. The moral disagreement is very high stakes, i.e. it's extremely important to both parties. Let's also suppose that the disagreement plays out in a situation where the outcome either favors one side or the other (so, not both, not neither).
Here's an example that I use to make things more concrete.
Suppose that you and your partner have a child that, because of some terrible circumstance, is now currently in a persistent vegetative state. The probability that the child recovers is very low. The two of disagree as to whether the child's life should be terminated. One of you wants to terminate, and thinks that it's morally permissible. The other thinks that it isn't morally permissible, and does not want to terminate the child's life.
Obviously this decision would matter, and the result will favor one individual, and not the other.
Suppose that you find yourself in a situation like this, and suppose also that you're an ethical relativist. How are you going to resolve this situation?
It seems that a situation like this will be resolved only through conflict, manipulation, or brute force.
Why is this?
Disagreements are resolved by appeal to some third party information or principle that all disagreeing parties recognize as authoritative. If Joe and I disagree about the score of last night's game, and if we both recognize espn.com as an authority, we can resolve our disagreement by checking the website.
But, if you're a relativist, and you're in a moral disagreement, what can you appeal to in order to resolve the disagreement? If you're a relativist of the subjectivist variety, then only authority presiding over moral is your own beliefs.
People don't think that relativism is a problem because they can ignore a lot of moral disagreement. All you need to do is hang out with like minded people and keep everyone else at arm's length. At that point, it's easy to be "tolerant" and to "agree to disagree."
Things aren't always so easy. Perhaps it is easy for individuals to avoid disagreement, but it becomes increasingly difficult for ethically diverse groups living in the same space to avoid moral disagreements. What are you going to do, then?
Consider the civil rights movement of the 1960s. You had two groups living in the same space that disagreed about how one of those groups should be treated. This disagreement mattered, and you couldn't just walk away from it. What are you going to do? If you're a relativist, you can't try to appeal to some greater moral principle to resolve the disagreement. This was the approach that Martin Luther King, Jr. took. This conciliatory approach requires some form of ethical objectivism, and of course King was no relativist about ethics.
So, what do you do if you're not an objectivist about morals? You fight until you either win or die, I suppose.