Had a friend just tell me that he had a run in with a game warden back in the day regarding a mercy shot on a downed antelope. He's saying that the only pistol calibers legal to fire that shot were .41, .41mag, .44, and .44mag.

I'm finding that hard to believe, given how many fudds do it with a .22.

Current Fish and Wildlife rules only list characteristics for centerfire cartridges used in the hunting of game. Does this mean that you have to use the same caliber on a wounded animal that's down as you do for one that's still running around?

Never crossed my mind ever if it needed done.

It is my understanding that Wyoming hunting regulations require you to use a legal hunting cartridge to shoot big or trophy game in Wyoming.  It does not matter if it is the first shot taken or the last.
 
In fact, it is my understanding, if you are carrying a .22 pistol along with your high-power hunting rifle while hunting big game, the pistol is prima-facie evidence of poaching; unless you are also carrying a small game license and can legally hunt small game at the same time.

But I have not attempted to stay current on Wyoming’s Game & Fish regulations since 2005, so the regulations may have changed.

I just re-read your question, the response above is based on a mercy shot taken after wounding an animal while legally hunting.  If it is an animal that is starving or has been hit by a car, only a Game Warden may legally take a mercy shot in Wyoming.