Mathematicians often treat mathematical structures as if they already exist in some infinite Platonic space, and they are just discovering them instead of creating them. A few ideas need to be disentangled here. The ideas don't "already" exist before someone discovers them, or "come into existence" after someone discovers them. The Platonic realm is without time, so neither of these ideas could apply to it. Also, the Platonic realm isn't some place; place is also something that doesn't exactly apply. What I mean by "exist" here is somewhat stretched from the usual meaning. What remains is the fact that independent observers, regardless of culture, when exploring the same mathematical structures, will discover the same facts about them. These facts are independent of the individuals involved. If you don't want to call that "existing" that's fine, but it is a lot like reality. It resembles reality in that way.
I think we can argue for the objective "existence" of Justice or Ethics in the same way. To the extent that observers can discover facts independently about the state of affairs, and these discoveries will agree when they overlap and fail to contradict, the qualities listed above have objective, mathematical type existence.
On this view, moral arguments are always resolvable if certain conditions are met. When a disagreement persists, the problem must be a definitional one, failure of one or both parties to follow through moral reasoning to a conclusion, or ultimately assuming certain moral statements which the other person would disagree with.