Generals don't resign. Generals soldier on. It's the tradition, it's the custom, it's the value. You check your conscience at the door, which is where you also check your politics. (The new professionalism of a 'disciplined disobedience' merits its own focus.)
Argue yes. Argue inside the four walls of the room and maybe pound the table once or twice. Few generals or admirals are ambiguous in the closed door discussions. However, even fewer flag officers ever collect their marbles and go home. You rather soldier on. It's what generals do in a republic whose first general was George Washington, where democracy is always being explored and developed and in which the civilian authority commands the military. Generals are nothing if they are not Americans first and foremost -- above all else. So generals obey which is their deep rooted tradition.
Indeed, since 1775 how many generals have resigned in protest of orders or policy? Few if any generals resign in protest. It is rare. Generals don't quit or throw in the towel because they might oppose going to war or in opposition to a peace treaty that ended the war. The same is true during wars. The WW II generals did not resign because the D-Day invasion of Nazi occupied Europe got moved to mid 1944 instead of the early 1943 the generals and admirals wanted. U.S. commanders in Vietnam didn't resign because Potus would not invade North Vietnam or for any combination of reasons Washington would not wage all out war. Rather, the generals say, "Yes sir, even if it's only because you said so sir." Generals who don't say yes sir get fired. Characteristically they go quietly and they fade away. Generals don't play sore losers either.
What is significant to the present is that in 1945 virtually all the generals and admirals opposed using the atomic bomb against Japan. This is fact despite Japan being Americans hated enemy of WW II. Our generals firebombed with minimum mercy the cities of Japan and our armed forces took it with feeling to the Japanese from Australia to Iwo Jima and Okinawa. We were going to invade the home islands to torch their fields and maul 'em all. Every last one of 'em if need be.
Yet the generals and admirals were against using the bomb. Using it. Dropping it. Launching it and exploding it on someone -- anyone. The generals and admirals are still against using the bomb. They just don't consider the bomb to be a weapon of war. The bomb is rather the destroyer of civilization itself. Using nuclear bombs leaves nobody around after it to sign a peace treaty. It leaves nobody around period. Using the bomb is not warfare. It is, in the words of the A-Bomb lead scientist Robert Oppenheimer, the destroyer of worlds. There's nothing military about it.
Indeed, the generals consider that using the bomb again would be madness. Only a madman would do it. And the generals know who are the few generals who would say 'yes sir' to execute a first use of nuclear bombs since 1945. As the general retired in the OP implies, there is only one military reply to a first use order and it is no. Not 'no' as a new overall and overarching policy but rather 'no' to the present circumstance in its specifics and particulars. It is this side of the equation the generals can influence or impact directly and immediately. If the other guy on the other end of the nuclear equation might make the first move then he should know already he'll fry in hell for it. And that the world will suddenly become a radically threatened place. That's not where generals or the people and societies they have pledged to defend want to be.
