We do not know how long after the main body of the movie that epilogue takes place. The exact time frame is left deliberately ambiguous.
Even when Kirk says it may be the only way to earn peace with the Romulans, Spock chooses revenge.
I'm sorry, but that's akin to saying that if the United States Army is in a firefight with Osama bin Ladin and he's about to escape unless they shoot him in the back, they shouldn't shoot. It's just absurd.
Yes, capturing him (be it Nero or bin Ladin) alive is preferable, but if that's not possible in combat, the logical thing is to protect innocent life by killing him.
Seeing as Kirk still has the scars on his face it would appear to be a few days at most. Spock goes through this horrible trauma and in less than a week is all better? He removed himself from duty and apparently declared himself fit again as well. Where was McCoy?
Nero wasn't going anywhere. The Narada was falling apart, pieces were falling off. It was on both sides of the wormhole, not going in one side and coming out the other in some other reality like we saw earlier.
This is akin to capturing an enemy solder who cannot run away, who has no weapons left and shooting him in the back. It's called revenge.
It's not surprising the support Spock has for his actions. How many war crimes have gone unpunished in Iraq and Afghanistan? If you take on the same actions of your foes, you become them.