Kirk addresses their penchant for argument in the message, but even if McCoy was prone to outbursts, Spock was not. So there would be no "mortal combat" to be locked in: those are the key words "locked in mortal combat." That's not these guys.
Spock would just shut it down. Like he did with Scotty and McCoy in "The Gamesters of Triskelion" by mentioning mutiny. Or he'd just respond logically and usually McCoy would cool back down.
Theses guys aren't schoolyard kids, they're adults and one of them is a ranking officer. It doesn't make sense that without Kirk the two of them would just figuratively start throwing hands because "Daddy" wasn't there to control them.
I can see the second choice, but I've always only ever heard it as the first, that Spock and McCoy were liable to be fighting a lot. After all, Kirk couldn't know that he would be killed while they were in combat. Maybe Kirk dies as a result of slipping on a banana peel during shore leave.
I know you're joking but really, Kirk would have stipulated that they run the orders if he is lost in a combat situation because that would be the most logical time for Spock to officially take command.
Here's the actual message:
KIRK [on monitor]: Bones, Spock. since you are playing this tape, we will assume that I am dead, that the tactical situation is critical, and both of you are locked in mortal combat. (emphasis mine)
"The tactical situation is critical" isn't "you boys is fightin'." It means the ship is in trouble and Kirk is dead. They are locked in mortal combat with whatever put the ship in danger. Not each other.
KIRK [on monitor]:
It means, Spock, that you have control of the ship and are probably making the most difficult decisions of your career. I can offer only one small piece of advice, for whatever it's worth. Use every scrap of knowledge and logic you have to save the ship (emphasis mine). But temper your judgment with intuitive insight. I believe you have those qualities, but if you can't find them in yourself, seek out McCoy. Ask his advice. And if you find it sound, take it.
Underlined: "to save the ship." Again, the ship is in danger, Kirk is dead, Spock has to step in. So far, Kirk isn't saying "stop fighting with McCoy long enough to ask him something." He's asking Spock to reach out to the other person Kirk trusts the most.
KIRK [on monitor]:
Bones, you've heard what I've just told Spock. Help him if you can. But remember he is the Captain. His decisions must be followed without question (emphasis mine). You might find that he is capable of human insight and human error. They are most difficult to defend, but you will find that he is deserving of the same loyalty and confidence each of you have given me.
The bolded is where Kirk addresses their habitual bickering. He's asking Bones - not Spock - to keep calm, to accept Spock's command and assist as needed, to treat Spock in the same way he treated Kirk. Because he knows McCoy would be too raw to fully control his temper.
Take the focus off
both of you are locked in mortal combat and put it on the rest of the message
. Granted, Kirk should have said "the
ship is locked in mortal combat" but he didn't. He did, however, say the rest of it.