Ugh. Where to start...

One might well ask: why does any of this indicate that moral propositions have no rational justification? The arguments presented here show fairly conclusively that our moral judgements are instinctive, subconscious, evolved features. Evolution gave them to us.

Yes, because evolution gave us the instincts that solved the prisoner's dilemma and made social life possible. Which is why Jonathan Haidt finds it more helpful to define morality as, rather than being about harm and fairness, something like:

Moral systems are interlocking sets of values, practices, institutions, and evolved psychological mechanisms that work together to suppress or regulate selfishness and make social life possible.

Green is basically screaming bloody murder at how people stupidly conclude that incest is wrong in a case where some bad attributes of incest don't apply, and how this is part of a more general flaw involving people doing an end-run around the usual need to find rational reasons for their moral judgments.

His view is in complete ignorance of recent ground-breaking research on the nature of human morality (see above link). Basically, most secular academics think of morality only in terms of harms and fairness, but, worldwide, people judge morality on three other dimensions as well: ingroup/loyalty (do we maintain a cohesive group?), authority/respect, and purity/sancity (the last one being the intuition challenged by Greene's example).

While political discourse in the West has focused on harm and fairness, human nature in general is judging from all five. This narrow focus has resulted in Westerners, not surprisingly, being unable to justify from the three others unless they come from a ... religious background!

Or, more succinctly, morality is a meme that enables solutions to the prisoner's dilemma. All five dimensions, to some extent, work toward that end.

What Greene has discovered is better described as "Westerners do not have the educational background to justify and express their moral intuitions that go beyond harm and fairness." Congratulations: when you force people to talk about morality purely in terms of harms, you can get them to voice moral opinions they can't justify.

Had the participants gotten such grounding, they could have answered the incest dilemma like this:

"As stipulated, there is no harm from what the siblings did. However, that's just disgusting [sanctity], and is disruptive to the social order [authority]. Within your artificial scenario, you have assumed these difficulties away. If I and others did not find such acts disgusting, they would inevitably become more common, and social life would break down: first, from genetic disease, and second, from destabilized family units, where parents are forced to take sides between their own kids. Over time, this hurts society's ability to solve the prisoner's dilemma."

The fact that people cannot connect their moral intuitions to the evolutionary/historical reason that such intuitions evolved is not a reason to come to the conclusion Greene does.

I should also add that it starts off with a very questionable claim:

As a simple example, consider the use of the words in any standard ethical debate - "abortion is murder", "animal suffering is just as bad as human suffering" - these terms seem to refer to objective facts; "abortion is murder" sounds rather like "water is a solvent!".

If you're correctly paraphrasing Greene, this is misleading at best. Yes, those statements are syntactically similar, but most people are capable of recognizing when a statement starts to make a moral claim (or, upon further questioning, some concept they hold that is isomorphic to morality). They recognize that when you get into talk about something being "just as bad" as something else, you're talking about morality.

It's like saying, hey, "Jews are murderworthy" sounds rather like "apples are red", OBVIOUSLY we are ill-equipped to discuss morality!

FWIW, I don't even necessarily disagree with Greene that people approach morality from a flawed framework. But his arguments aren't very good, they ignore the literature, and don't present the right framework. Thumbs down.