I am not that familiar with the period, but I doubt very much that an officer/ captain on board ship in battle would have had the first clue what to do with an injured man, from a medical viewpoint, other than to call the surgeon and let him deal with it. (Women, on the other hand, seem to have had some experience with fainting and always had smelling salts.) The officers would have been busy with maneuver and firing of weapons, which wouldn't be of much use in the current situation. Perhaps CW should have thought to call a surgeon, but as has been pointed out, this was not his element. Decisions that are immediately obvious to us in a familiar context can be completely perplexing in a different situation. On board a ship he would have known his surgeon by name and character; in Lyme he doesn't know who to call or where he is.

Also, I think FW is momentarily paralyzed because he does feel responsible for what happened, and as has been pointed out, this is not a situation of his sailors who have volunteered (more or less) for naval duty and know the risks. He certainly would have been familiar with family notification and all the attendant distress. We hear of this in Harville's account of how CW told Benwick of Fanny's death, and we saw it previously when he talked about Dick Musgrove with Mrs. M. I don't know whether it might have consciously crossed his mind at the moment (though I bet it would at least be a subconscious thought) --that Mr. and Mrs. Musgrove might now have lost two children.

And...men seem to react more strongly when women are injured/hurt than when men are, presumably because men are/were conditioned to be protectors of women. (Presumably they can be/are trained out of this distinction in the modern military setting, where it can be disruptive to good order.)

I don't see this at all as a PTSD (post traumatic stress disorder) type of situation, though.