Alfred North Whitehead once famously said, "The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato." To that end, much ink has been spilled over Plato's Euthyphro dialogue. You can read it here. The very simple point I will try to make is that what Plato was gesturing toward in Euthyphro is exactly what classical theists mean when they say ‘God’. But more importantly, the classical theist understanding of God as Pure Being/Act (as opposed to God being a Person with maximally great properties, etc.) shows the futility in using Euthyphro to indict classical theism.
As the reader is more than likely aware, the dialogue builds up to a dilemma where Socrates asks "Is the pious loved by the gods because it is pious, or is it pious because it is loved by the gods?" The question aims at the definition of piety, and the dialogue ultimately ends in an impasse. Euthyphro gets frustrated with Socrates’ prodding and heads off to piously prosecute his father.
In the millennia since Plato, those unfriendly to theism have recast Socrates' question in the following way "is something good because God wills it, or does God will something because it is good?" If the theist takes the first horn, it seems that what is 'good' is arbitrary. On this line of thinking, God could will abominable acts which would technically be ‘good’. If the theist takes the second horn, then there is something beyond God which even He is subject to, by which God says something is good, and thus God is not sovereign and/or metaphysically supreme. Thus, taking either horn impugns the nature of God and sinks the theist.
We should note that the theistic dilemma differs somewhat from what Socrates and Euthyphro discuss. If we just take the term 'piety', then it seems that we are talking about defining a virtue (or virtue itself). In this case, we might come to Euthyphro's aid and present a better definition. But what Socrates is really asking is “what grounds piety?” Certain actions might be pious, but why is this so? What is the objective means by which we say that something is pious?
It seems obvious that Plato answers the Euthyphro dilemma in his later dialogues. When we learn about the Forms we can surmise the pious by Pious/Piety. And Piety is ultimately rolled up under the Good. Temporally pious actions or a pious person exemplifies or participates (and so forth) in Piety. Piety is the objective means by which we say something is pious or impious. So, perhaps Socrates would have been satisfied if Euthyphro would have said: "something is pious by the Pious” and gone on to explain the Forms. Such would have been sufficient to show Socrates that the dilemma was indeed false.
Theists typically respond to the Euthyphro dilemma as being false when this objection is leveled. The theist tries to split the horns of the dilemma; X is good because God is good. Goodness, Piety, and so forth are grounded in God’s nature. Thus, there are no arbitrary goods and nothing beyond God which He is subject to. God is the ground of objective moral values and duties. Or, so the typical reply goes.
The question then becomes how is God the ground of moral values and duties. This seems to be a fair question and demonstrates why I think the classical view of God decisively splits the horns of Euthyphro. It is because the classical theist conceives of God as absolute metaphysical Goodness. God is Pure Being/Act, having all perfections infinitely and simply, and is the only thing that could truly fulfill Plato’s notion of the Good. I think other theistic conceptions, most of which fit under the guise of ‘theistic personalism’ (read more about that here) face difficulty in splitting Euthyphro’s horns. This is because, on theistic personalism, God is a being that has the property of goodness; God is called good because He has the property of goodness to a maximal degree.
But saying that something has goodness versus that something is goodness are two very different propositions. The classical theist directly appeals, via analogical predication, to the nature of God as goodness itself, whereas the theistic personalist must say that God is a being that has goodness. On the theistic personalist view, it is difficult to think of how God is not metaphysically composed in some way, being made up of an amalgamation of maximally great properties. And if God’s composition includes the property of goodness, one cannot help but perceive this property existing apart from God. And if goodness can, in principle, exist apart from God, then we are right back on one of Euthyphro’s horns.
The Euthyphro dilemma ultimately has no force against the classical theist. The notion of the Good that Euthyphro alludes to is fully and truly explicated in the work of the great classical theistic thinkers, such as Thomas Aquinas and those following in that scholastic tradition. These thinkers would, I believe, be unanimous in agreement that nothing composite could be the metaphysical absolute capable of splitting Euthyphro. And I think they would also agree that the dilemma is clearly false once a robust metaphysical inquiry is undertaken.
It is beyond the scope here to show how an atheistic Platonism would fail, and I may revisit this question later. I will simply state for now that Pure Being/Act is the only possible metaphysical Goodness per se. Plato’s system cannot provide a robust conception of being (), and such a notion is necessary to have that which is Goodness per se.