The Natural Theist (@AleMartnezR1) on twitter posted that non-theist views of morality end up being subjective. As such, they cannot ground objective moral truths.

There are several problems with this comment, but I want to focus on a reply to Natural Theist by The Real Atheology Podcast (@RealAtheology):

Real Atheology includes an excerpt of Oppy's opening statement (§27) in Is There a God? A Debate. Oppy claims that moral principles are true because they are necessarily true. They are akin to mathematical truths. Just as it's necessarily true that 2+3=5, it is necessarily true that killing is immoral sans special circumstances, such as self-defense and justified military and police use of deadly force. It's not possible for such moral truths to be false. 

Oppy's view on morality is in opposition to proposition T10 defended by Kenny Pearce, 'There are objective moral facts because God legislates them'. Oppy concludes: 

I say that there are objective moral facts because there must be. And I say that there is no explanation of why there must be objective moral facts. I think that there are independent difficulties for the view that there are objective moral facts because God legislates them. But these are not my primary reasons for rejecting T10. (emphasis mine)

For Oppy, fundamental moral principles are true of brute necessity. There's nothing further that could explain why they're true as they don't depend on anything to ground their truth.

Pearce points out that Oppy's view cuts against how we normally do moral reasoning:

Graham holds that there can be no explanations of moral principles (see §27 of his opening statement). However, it is commonplace to appeal to more general moral principles to explain less general moral principles. One of Graham’s examples of a moral principle is, “It is morally wrong to torture babies for fun.” Why is this true? Because it’s morally wrong to torture anyone for fun. Why is that true? Because it’s morally wrong to take pleasure in causing suffering to others. Why is that true? Because suffering is bad. There are many philosophical difficulties in spelling out the details of these explanations, including the difficulty of understanding precisely how wrongness relates to badness in the last stage of the explanation....Graham’s denial that moral principles ever stand in explanatory relations to one another isn’t just throwing out moral theory, it’s also throwing out a large part of our ordinary reasoning about what we ought to do. We often can and do give explanations of moral principles.

Now you might wonder, why does Oppy hold this view? Pearce summarizes how Oppy arrives at this view regarding the necessity of mathematical truths:

According to Graham, every non-modalized claim (every claim that does not contain modal operators like ‘possibly’ or ‘necessarily’) has an explanation, and these explanations are of two kinds. If a claim is contingent, then we explain it by pointing to something on which it depends. If a claim is necessary then we explain it by pointing out that it is necessary. So, Graham says, we can explain why 2 + 2 = 4 by pointing out that it is necessary that 2 + 2 = 4. However, according to Graham, “there is nothing that explains why it is necessary that 2 + 2 = 4” (§4).

A further cost of Oppy's stance on morality is that it results in infinite commitments, which counts against his brand of naturalism having the theoretical virtue of simplicity. As Pearce explains:

According to Graham, all necessities are brute. This view does not just run contrary to intuition; it also runs contrary to the practice of moral reasoning and mathematical investigation. Furthermore, in the game of worldview comparison, the costs of Graham’s principle of brute necessity are literally infinite. Graham holds this view in part because he denies that one necessary truth can depend on another. This claim about dependency relations is implausible. Once we reject it, we can see that classical theism can provide an attractive and highly unified picture of the dependence of necessary truths on God.

Pearce argues that God grounds objective moral truths, and he thereby provides a response to the Euthyphro Dilemma:

all of the necessary truths have their origin within God, yet insofar as God is a rational being it is impossible that God should have made different necessary truths. God values or endorses the laws of ethics, logic, mathematics, and so on in such a way that, necessarily, God chooses in accord with them.

Yet because God’s choices are guided by God’s values, this limitation on God’s possibilities for choosing should not be understood as a constraint that diminishes God’s freedom (see Pearce and Pruss 2012, 410–412).The necessary truths arise from God’s nature as a rational being. For this reason it is true, on my view, that if God were irrational, there might be true contradictions (Pearce 2017a, 8). But necessarily God is perfectly rational, and necessarily there are no true contradictions.

How precisely do the necessary truths arise from God’s nature? The general picture is that, in making a free and rational decision about whether and how to create, God necessarily endorses the necessary truths as laws of thought that (like the moral law) God makes for Godself.

Pearce splits the horns of the Euthyphro Dilemma. Moral truths depend on God (i.e., God doesn't depend on a moral standard outside of himself), but moral truths are not arbitrary (i.e., they arise out of God's nature and they could not have been otherwise).

Without going down the rabbit hole of the Euthyphro Dilemma I want to show another problem Oppy's view has. His view is cast into doubt by an abductive version of the moral argument.

In Two Dozen (or so) Arguments for God, David Baggett discusses an abductive moral argument that runs:

7.There are objective moral obligations.

8.The best explanation of objective moral obligations is God.

9.Therefore, (probably) God exists.

Oppy endorses (7). So how would he argue against (8) to undermine the argument? More specifically, would his own brand of naturalism offer a better explanation of objective moral obligations in comparison to a brand of theism like the one Pearce puts forward?

Moral obligations arise out of the rightness and wrongness of actions. They focus on what we ought to do and ought not to do, given whether an act is right or wrong. We are morally required to not perform a wrong action, and we often are morally required to perform a right action (right actions can also be permissible, neither obligatory nor impermissible). 

Moral principles are general statements about what is right and wrong. For instance, Oppy offers the following as two true moral principles:

(1) Killing: it is morally wrong to kill, except in special circumstances.

(2) Torture: it is morally wrong to torture babies for fun.

Regarding (2), we always have a moral obligation not to torture babies for fun. Whereas, in most circumstances we have a moral obligation to not kill, as (1) indicates.  

Further, Oppy claims:

It is easy to extend the list of fundamental moral proscriptions: it is outright morally wrong to engage in sexual assault; it is morally wrong, except in special circumstances, to lie or steal; and so on.

It is also easy to make a list of moral prescriptions: it is outright morally required that we collectively make adequate provisions for the least fortunate among us; it is morally required, except in special circumstances, that we obey the laws of the land; and so forth.

With these pieces in place, what explains the truth of moral obligations for Oppy? Objective moral obligations hold because there are objective moral truths. What explains these truths? Nothing. They are brute facts. They're true of necessity, perhaps because they are features of a segment of the initial universe that exists of necessity. 

Oppy's explanation of objective moral obligations leaves much wanting. It's almost a non-explanation. As he says, "there is no explanation of why there must be objective moral facts." So, even if Oppy raises worries concerning a theistic explanation of morality like Pearce's, such an explanation is a better explanation of objective moral obligations than the minimal or non-explanation on offer from Oppy's worldview. Thus, an abductive moral argument like Baggett's is particularly damaging for Oppy's view. God is the best explanation of moral duties.

By contrast, Oppy's view has no problem handling deductive moral arguments. Consider this William Lane Craig-style argument mentioned by Baggett:

1.There are objective moral obligations.

2.If there are objective moral obligations, then God exists.

3.So, God exists.

The contrapositive of (2) is that if God doesn't exist, then there are no objective moral obligations. However, Oppy can easily counter this by claiming that moral truths are a necessary objective part of our universe and the obligations that fall out of them are likewise necessary. If there's no God, you can still have such obligations on an Oppy picture of reality.

Returning to where we started, Real Atheology included something else in the tweet reply to Natural Theist. 

Real Atheology includes a copy of Morriston's article in a book addressing Craig's debate over morality with Erik Wielenberg (A Debate on God and Morality). However, Morriston's chapter cuts ice against an admittedly flat-footed view of the objectivity of morality as that embraced by Craig. It applies to a deductive moral argument, but it doesn't impugn a nuanced view of moral obligations, such as the Kantian-inspired view embraced by Pearce. It also doesn't cut ice against an inductive moral argument. 

In summary, Real Atheology is correct to point out that there are other ways of grounding the necessity of objective moral truths other than God. However, Oppy's way of doing this by making moral truths necessary features of our universe runs into many problems that call it into question and make it a worse explanation of the objectivity of morality concerning moral principles and the obligations arising out of them.