One shot

"The Queen" has a deer scene, ring a bell?
"One shot" will take a life, for a deer, for a man, but not for The Terminator ;P

I feel like the other 2 people didn't really answer your question, so here's my attempt.

Towards the beginning, Mike mentions that you're a "pussy" if you can't kill the dear with one shot. I suppose in his mind, it goes to some kind of macho thing. Spoiler from here on...After the war and going through the horror of "one shot" that might have killed him or Nick in the RR scene, he seems to figure out that "one shot" isn't necessarily a good thing. When Nick takes his last shot, he says, "one shot" right before it- indicating that he DID (at last perhaps) recognize Mike, but all too late.

I thought Michael had had enough of killing - he sees the deer in its habitat - alive and in nature - and decides not to shoot. He sees no sport/enjoyment in killing now.
Right?

"Remember, you have to make it home to get paid" (The Dogs of War)

I thought part of the essence of the "one shot" them was that you kill the animal in one shot so it did not suffer too much. Whereas all these guys were tortured by playing Russian roulette, which was unnecessary punishment if they planned on killing them was inhumane. I think Mike not killing the deer was a sign that he just didn't want to kill anymore, but wasn't that reflective of the one shot theme. In the last scene, Nicks mental suffering ends by him taking one shot, a quick clean death. A very powerful movie that leaves much up to interpretation.

I think "one shot" was Mike's philosophy on life. He believed everybody got only one shot at life, so they'd better make the most of it. Note that he hits on his best friend's girlfriend at the wedding reception, because he's going to Vietnam, and feels he may not get another chance to let her know how he feels about her. Also, when they were P.O.W.s, remember what he said to Nicky, when he came up with the plan for escape: "Nicky, it's the only way." They only had one shot of getting out of there alive. There was also the line at the beginning of the movie, after he passed the truck 'on the inside': "There's only one shot against a sure thing." As far as why he couldn't kill the last deer, I think he knew what it felt like, at that point, to be on the other end of the barrel, and could no longer kill a defenseless creature.