Here's a nice summary of his argument:
We have an empirically confirmed theory about where our moral judgments come from (we are supposing). This theory [i] doesn't state or imply that they are true, [ii] it doesn't have has a background assumption that they are true, and, importantly, [iii] their truth is not surreptitiously buried in the theory by virtue of any form of moral naturalism. This amounts to the discovery that our moral beliefs are the product of a process that is entirely independent of their truth, which forces the recognition that we have no grounds one way or the other for maintaining those beliefs. (p.211; emphasis mine)I have italicized the inference that I think is mistaken.
Consider the following belief-forming mechanism: If I read something Stephen Hawking writes about astrophysics, then I believe it. Suppose the correct explanation (corresponding to 'empirically confirmed theory' above) of why I believe that there was a big bang is that Hawking wrote 'There was a big bang,' I read it, and that I have this belief-forming mechanism.
This explanation (/'theory') of why I have this belief
(i) 'doesn't state or imply that 'There was a big bang' is true';
(ii) 'doesn't have as a background assumption that my various Hawking-derived astrophysical beliefs are true'; and
(iii) the truth of 'There was a big bang' is not 'surreptitiously buried' anywhere in the explanation of why I have this belief.
And yet, this does not 'amount to the discovery that my Hawking-derived astronomical beliefs are entirely independent of their truth.' (Of course, you can insert any other genuine expert and area in for 'Hawking' and 'astrophysics,' if you think Hawking's writings in the field aren't truth-tracking.)
So Joyce needs more than (i-iii) to demonstrate that moral judgments are unreliable.