First let draw the distinction in what I feel and what I think Cap feels. I personally don't have a major problem with us dropping the bombs. But I'm an amoral jerk and I would have wanted payback for Pearl Harbor. But Cap isn't me. He has morals and values and ethics and a bunch of similar words that don't apply to me either.
While Cap certainly believes "war is hell" I do think he would have had a problem with his superiors ordering the destruction of Hiroshima (and THEN Nakasaki). As a modern comic reader I don't think modern Cap is the ends justify the means type character. But I grant it's war time and a different era where dissent is frowned upon so maybe he wouldn't actively oppose the bombings or snitch and tell the world about the government's plan...but I don't think he could stay Captain America after the bombings if he hadn't been frozen. I don't think he could live with it. He would feel there was probably another way (logical or not) and I think it would eat at him. I don't think he could still traipse around in a colorful costume espousing how great America was after we nuked hundreds of thousands.
Could the government have found another way like showing a demonstration of one of the bombs on an island for the Japanese leaders? Did Nagasaki have to be destroyed...if we gave the Japanese a week after Hiroshima would they have surrendered by then? Were the Japanese already considering surrender? Would Tokyo have been a better military target than say Nagasaki which I think was a fishing village? I think the what ifs would eat Steve alive and he would question his own patriotism.
And this is probably a very wide tangent to go off on because Cap would probably have no knowledge of it...but like I said in another thread the uranium used in the Manhattan Project was extracted from the Shinkolobwe mine in the Belgian Congo...it wasn't the Americans or Beglians workers who were used to extract that radioactive ore...it was the Congolese. This was already a bloody business before the Uranium was refined much less the bomb dropped. But like I said Cap probably couldn't have known about it so forget I mentioned it.
Maybe the Cap as written in the golden age could have supported the decision to nuke. He wasn't the version we have today who is a cliche that is for everybody. The modern version doesn't have that American exceptionalism thing going on and would never be racist towards the Japanese. Maybe Golden Age Cap would have been the ends justify the means type guy. I just can't see the modern Cap ever agreeing with the decision or taking pride in the cost of ending the war meant sacrificing that many lives. Ever since I've been reading him Cap isn't about moral relativism. Saving the most lives doesn't justify an atrocity that still kills numerous lives (this is where we differ...I very much do).
You have a better argument than I do. But from what we've seen of Steve in modern times I don't think he would be at peace with the decision to nuke Hiroshima and Nagasaki. And if he learned about the decision prior to the bombings I don't think he would have been able to handle it. As a soldier I don't think he had any conception of just how destructive a nuclear weapon could be and I think he would be horrified at what this new weapon could do in either Americans hands or one of our enemies (or even our allies). In my head if Steve had been around to hear about the bombings he would be a very depressed individual knowing his country had the potential to level cities with a bomb and that someone else was eventually going to get their hands on one. I don't think Steve would approve of nuclear weapons in general much less the way we used them. But that might be me misunderstanding the character.
If Cap WOULD have quit...then he is a hypocrite for not quitting soon after her unfroze.
If he would have felt so strongly as to quit or try and stop the bombings he should never have continued after he found out.
I think he would have hated its being used but also accepted its use. Moral or not he was a man of his time...today he might oppose such a thing directly...but not then. He might come to think we have to find a better way (and we do), but I can see him accepting its use.