Put simply, the modern version of the Euthyphro Dilemma is usually presented something like this:
Are morally good acts good by virtue of their own nature, or are morally good acts good because God says they are good?
The first horn of the 'dilemma' implies that the good is external to, and thus independent of, God. The second horn implies God's commands would, therefore, be arbitrary.
There are multiple problems with this. We'll list a few.
1. The Euthyphro Dilemma assumes a very low view of God. It assumes a non-specific God who hands down to a disconnected creation laws which He is either subject to by virtue of their already existing outside of Himself, or to which He is loosely related through His arbitrarily revealing them to the creation.
It is important to point out that God's commands, or divine laws, flow from His very nature, which is essentially good. Being the Creator and Sustainer of the universe, God is beholden to no-one and nothing outside of Himself. As necessary Being, we can say, No God, no good!
2. The God of Christianity is Triune. The Father, the Son and the Holy Spirit relate to one another necessarily and eternally. This interrelationship provides the very foundation of morality. The Persons of the Trinity are not beholden to any external law, nor are they subject to the arbitrary commands of one or the other. Rather, they are in co-relation out of perfect and uniform love for one another. God's commands, or laws, are a reflection of His very character and nature, not the result of whimsical arbitrariness or impulsiveness, nor are they the result of laws external to God to which He is beholden.
Once we take this into account, along with some of the essential attributes of God, like the supremacy of God, the sovereignty of God, the immutability, or unchanging nature of God, the self-sufficiency of God, and the goodness of God, we begin to understand that God's character and nature is the very standard of all that is good, and the objections posed by the Euthyphro Dilemma vanish. God loves morally good acts because He is good, and therefore His commands reflect His essential goodness. God is entirely self-sufficient, and is in need of nothing outside of Himself.
3. In some sense it is true that God loves morally good acts because they are morally good, and in another sense it is true that morally good acts are that which God commands. But this is a mere tautology. A necessary truth. It does not entail that there is a standard outside of God, nor that God's standard is arbitrary, and to argue such is to offer an incomplete analysis.
We have an innate awareness of God's divine commands, or laws. (Romans 2:15), thus moral obligations are divine laws. There is a necessary relationship between God's moral law and our moral obligations. Duty-related properties depend on God's commands, but evaluative properties, such as goodness, do not.
It is true that an action is morally obligatory since God has commanded it, but the goodness of an action does not depend on God's commanding it; the goodness itself flows from God's essentially good nature.
The proponent of the Euthyphro Dilemma usually fails to take into account this distinction.
Now, a standard objection will look like this (or some variation thereof):
'So God could have commanded that rape is good?'
No. God's very character and nature would prevent Him from doing so. See the non-arbitrariness of God's commands above.
1. To repeat, this simply ignores the rational and valid explanation given by the Christian, and is a rather transparent attempt to save the dilemma. God's very character and nature would prevent Him from declaring rape a morally good act. (see above.)
2. The objection shows that the objector is aware that rape is in fact not a morally good act. The contrast is clear. The objector attempts to communicate that God 'could' have commanded something bad to be good, hence the objector, in the very objection, demonstrates that they have an innate knowledge of what is good and bad. The objection demonstrates they are acutely aware of the absurdity of declaring rape to be a morally good act. And if they are aware of this, how much more God?
Soli Deo Gloria