[How] does the Church address the Euthyphro Dilemma?
We're getting well into stuff that I don't really know here, but as I understand it, classical Catholic moral theory bases moral truths not upon God's will but upon his nature. So truth, justice, etc. are good because they reflect the divine nature, which is in some sense Truth, Justice, etc. The Euthyphro Dilemma, at least as it's normally constructed, is really directed against forms of Divine Command Theory, which bases moral truths not upon God's nature but upon his will, i.e. telling the truth is good because God commands it, and so on. This is vulnerable to the objection that if God has a reason for commanding truth-telling rather than lying, then that reason is what really makes the one good and the other bad; whereas if he doesn't have any such reason, there's nothing intrinsically good about telling the truth or bad about lying, and God is just whimsical in his commands. If, however, the goodness of truth-telling rests ultimately in God's nature rather than his commands, then truth-telling is necessarily good (because God's nature couldn't be other than it is) and there is a reason for this, but the reason doesn't lie outside God. So, as I understand it, a divine nature account of moral truths isn't really susceptible to Euthyphro, at least as usually put. Of course that doesn't mean it doesn't face other objections.