In the case of gay sex and abortions, the disgust is the person's emotional response to something that isn't the most important (to the image provider, and perhaps to the providee too) part of the action. Dicks in butts (because anti-gay people always talk about gay men, never lesbians (probably because if they did they wouldn't get the desired response)) are less important to them than adults being able to do something they really want to do with their own bodies. Tiny bloody fetuses having been killed aren't as important as people's lives not being ruined by children they can't support, or people dying in childbirth who could have been saved by abortion.
On the other hand, piglets getting their testicles ripped out without anesthetic (and similar bad things that happens to animals) are the most important thing about eating meat.
Retracted cause this doesn't really work for the abortion example. It does for the gay sex one though. For the abortion one, I guess I won't condemn abortion based on some universal principle of how arguments should be conducted.
I notice that a large fraction of Effective Altruism people are vegetarian. This makes sense: in general Effective Altruists take moral issues seriously, even when that means changing your lifestyle. I'm not sure it's a good balance, though.
One way to think about this is to convert it into money. How much would I need to be paid to give up eating meat? All animal products? How much money would I need to spend on myself to be about as happy as I would be with less money but continuing to eat animals? I'd probably be willing to go vegetarian for about $500/year, vegan for maybe $2000/year.
It turns out you can probably pay to convince other people to be vegetarian much more cheaply than that. I estimate the cost of a vegetarian-year at $4.29 to $536 while Brian Tomasik estimates $11 with better methodology (which I looked at). This is by placing ads on facebook for a site where people can watch an animial cruelty video and ideally become vegetarian or vegan.
If you would get more than $11/year worth of enjoyment out of continuing to eat meat, why not give $11/year to convince someone else to not eat meat for you? Or give $50/year and be on the safe side?
(While you're giving money, you should probably give it to the organization that you think will do the most good with it, which I think is probably one of GiveWell's top charities. The nice thing about money as opposed to actions is that it's easy to redirect.)
I also posted this on my blog.