Depends on the fighter, I’d say. I can’t imagine why not though. Unless he sees it plucked from his skull with his other eye, he might just think it is damaged and continue fighting. Sometimes in a fight, blood gets in your eye and you can’t see. Doesn’t stop any decent fighter. Hinders them, perhaps, but never stops them.
I think you (general you), VASTLY overestimate the training of a Special OP person. Very little hand to hand is going on. Learning to defend one self is important, but is pretty trivialized by the US military. You are training an army of soldiers, (and yes, even the special ops) not an army of fighters. Solders use weapons and strategy, not skill at hand to hand fighting, and thus, spending hours and hours training how to kill with your bare hands is a waste of time and money for the government.
Special OPs are trained to endure special circumstances, to be able to take a lot of abuse and still keep on ticking, to use any weapon they can and many other various tasks that DO NOT include some uber bad ass hand to hand fighting.
A UFC person fighting a Marine Recon would more than likely win if the fight is hand to hand. Give the Marine a gun, a mission and throw a hundred UFC fighters at him and he will win, sure, hand to hand… No way.
Sorry, your supposition that Special ops spend all their time training to be ultra bad ass ninja killers makes me think you have never been in the Military. I could be wrong, but if so I think you have been misled.