Clarifying the question: how to know the good
Your question reflects the Euthyphro's dilemma as adapted by Christian philosophers / theologians, turning the question from the nature of piety & loved by God into a question on the nature of goodness & God's command. In both the Greek & Christian versions, the existence of God is not at issue, but on how to define piety / goodness in relation to God.
Euthyphro's dilemma has more to do with how we know the good, an epistemological concern: can it be known objectively in human nature without resorting to what only God knows?
One side of the dilemma answers YES, the position of moral realists, where there is an objective morality based on nature as opposed to socially constructed or individually constructed (subjective) morality. In the hands of Christian philosophers, this objective morality is renamed natural law to distinguish it from other laws in a hierarchy: eternal law, divine law + natural law, human law. This position is held by many Anglicans and Catholics including C.S. Lewis (see Problem of Pain, chapter 2).
The other side of the dilemma answers NO, where what God says is critically necessary to our understanding of piety or goodness. In the hands of Christian philosophers, this position is called the Divine Command Theory (theological voluntarism) held by many Protestants (especially Lutherans and Calvinists). This camp also does NOT believe in subjective morality but the objectivity is construed differently, basing it on God's commands rather than human nature. For this camp there is no natural law since it is conflated into divine law.
This epistemological issue of how to know the good is separate from how conscience informs us of what is right and wrong. Christians in both camps agree that our soul has conscience put there by virtue of our being created in God's image. Our conscience also operates similarly for both camps in that conscience is asked to acquiesce freely to a moral demand which ultimately comes from God.
The difference lies NOT in the operation of conscience but when Christians try to reflect philosophically on the question: how is goodness related to God's command? Can we enrich the concept of goodness apart from:
- what we discover in our conscience (through the moral demands)
- what we learn about God's nature from revelation (the Bible)
The YES camp (God command something because it is good) says: "yes, there is an additional source, through human nature (ontology) by virtue of the unity of the three transcendentals of Truth, Beauty, and Goodness." For the YES camp this brings enrichment and stability to our concept of goodness, where limiting our understanding of goodness to the limited information provided in the Bible contributes to the sense of arbitrariness. In the hands of Christian philosophers all 3 sources are harmonized, giving priority to revelation.
Can we have objective morals without God?
Supporters of natural law will say yes, and this can be the basis for well-meaning citizens constructing human laws cooperatively on the basis of objectively known human nature.
But Christian natural law supporters (such as Catholics) will immediately point out that natural law can take one only so far, to the level of pagan wisdom such as Aristotle and Plato. Christian goodness, happiness, and virtues require more than practicing Aristotle's Nicomachean Ethics. There are at least 2 areas where natural law on its own is not enough for a Christian:
We need the Holy Spirit who infuses us with the theological virtues of Faith, Hope, and Love that enable us to fulfill our inborn yearning for God as well as to enable us (by grace) to obey that law perfectly. Even in the natural realm, we need the Holy Spirit to infuse us with the four cardinal virtues of Prudence, Justice, Temperance, and Fortitude:
Natural law doesn't point us to our supernatural destiny so that the choices made by saints, martyrs, and people who renounce worldly riches can be understood as rational. Only Christianity teaches that we are made for another world where there is a resurrection into a glorious body in a new heaven and earth. Without that knowledge it doesn't make sense to choose first the Kingdom of God.
Does the natural law exist apart from God? Or is it from God?
If we are happy to stop at the natural law level to attain the objective meaning of goodness (within the natural realm) when discussing ethics with non-Christians, agnostics and atheists, then yes, it can function apart from God.
But if people push further into the PURPOSE (telos) question or the ORIGIN question, we need God to explain why this natural law is embedded into our conscience and why we have conscience in the first place.
If we want answer to HOW to fulfill the demands of natural law (in light of our sinful nature which continually defeats us) we need teachings on Salvation and Jesus.
If we want answer to "Is natural goodness is all there is to life?", which is a MEANING OF LIFE question, then we need teachings on Resurrection, Heaven, and the Kingdom of God.