"'What that fellow did was morally unacceptable. It was wrong' . . . Both of these expressions claim to tell us moral facts about the action."
No, they don't. That's just a way to interpret it, where the interpretation would hinge on a realist/objectivist misunderstanding of what's going on with moral utterances.
The following comment won't make a lot of sense to those who didn't read the article but:
What tilts the scale? A feeling! An uncomfortable feeling that I ought to go, just in case she really needs me. How do I account for this uncomfortable feeling? It seems that there is a belief, right at the beginning of my inner debate, that I ought to fulfil some filial obligation; a belief based on an assumption of some moral fact about parent/child relationships and that this belief plays a role that is not accounted for in Hume’s theory. The important thing about this belief is that it destroys the simplicity of Hume’s model (Desire > Reason > Action), by appearing as an additional feature at the beginning of the sequence.
Here the author is being quite careless about the feeling/belief distinction he just set out. He's calling the same thing both a feeling and a belief here.
Re Hume's analysis, "fulfilling some filial obligation" is as the author first recognizes, a FEELING that he has; it's a DESIRE on Hume's terms. A desire to fulfill a filial obligation he feels. (The Humean analysis isn't how I'd put it; I'd put it in different terms re it being a feeling or disposition that he has; I don't think analyzing all dispositions as "desires" really works. But the overarching point is that there is nothing cognitive/objective about this.)
I am suggesting that there are beliefs which arouse our feelings
You might believe that your FEELING of familial obligation is something other than a personal feeling you (and those who share it) have, but there's zero evidence of that.
namely that we sometimes desire to do something, because we already think that it is right,
Your "thinking that x is morally right" is a feeling, a disposition you have. There's no moral fact to that aside from the fact that it's a moral stance that you feel.
If a moral realist believes a certain action to be right, he might say: “I do not call the action right because I feel in a certain way. I feel in this way because I think the action right.”
They can say and believe whatever they want. The truth is that "thinking the action is right" isn't anything other than a feeling or disposition they have. Again, there's ZERO evidence that there's anything more to it than that. That the person has a belief otherwise is no evidence.
Professor Jonathan Dancy puts it like this:
“When I am standing on the kerb looking for a gap in the traffic so that I can cross the street safely, I am not doing this because I desire a long and healthy life. I experience no desire; I m just looking for a gap in the traffic before I cross. Why insist that there must have been a desire in there somewhere?
Not that that example has anything to do with morality, but actually, if the motivation for doing it has nothing to do with any sort of desire or emotional disposition of preference Dancy has, it would be because by this point it has become mere habit. Historically, if we go back far enough to why Dancy first started doing this, there would be some desire/preference involved, which might have simply been something like "a preference to not be scolded by mum for not watching out for traffic well enough."