Why does Nick say nothing to Mike except for "one shot"?

For understand "One shot" you have to go back to his early discussion with his friends including Michael:

Michael: I'll tell ya one thing, if I found out my life had to end up in the mountains, it'd be all right, but it has to be in your mind.

Nick: What? One shot?

Michael: Two is pussy.

Nick: I don't think about one shot that much any more, Mike.

Michael: You have to think about one shot. One shot is what it's all about. The deer has to be taken with one shot. - wikiquote

That's why he said "One Shot" as that's what it takes to kill deer.

Michael: I came 12,000 miles back here to get you...What's the matter with you? Don't you recognize me?...Nicky, I love you, you're my friend. What are you doing? We don't have much time, Nick. [Nick pulls the trigger on a gun, clicking on an empty chamber] Is this what you want? Is this what you want? I love you, Nick. [Michael pulls the trigger, clicking on an empty chamber] Come on, Nicky, come home. Just come home. Home. Talk to me. [looking at Nick's track marks] What did you do to your arms? Do you remember the trees? Do you remember all the different ways of the trees? Do you remember that? Do you remember? Huh? The mountains? Do you remember all that?

Nick: One shot. [He smiles and laughs in recognition]

Michael: One shot, one shot.

[Nick pulls the trigger, shooting himself]

Michael: Nicky, Nicky, don't, Nick, no!!

So he did remember him in his last memories but he was so broke inside that he still killed himself. He was dead inside already.

Why he can't recognize him can be due to Dissociative Amnesia

Dissociative amnesia is a condition in which a person cannot remember important information about his or her life. This forgetting may be limited to certain specific areas (thematic), or may include much of the person’s life history and/or identity (general).

Dissociative amnesia has been linked to overwhelming stress, which may be caused by traumatic events such as war, abuse, accidents, or disasters. The person may have suffered the trauma or just witnessed it.

Or some other memory disorder but we can just guess as it was never explicitly told.

The concept of "One shot", as taken in context during the conversation in the beginning of the movie, indicates a clean and humane way to hunt a deer. One shot means no suffering. Pull the trigger, the deer is dead.

At the end, when Nick says, "One shot", he's acknowledging Mike but also acknowledging that it's the only way to end his own suffering from all the horrors of war that he was exposed to. The suffering from PTSD. Pull the trigger, he's dead. Suffering ends.

My interpretation is that Nicky felt devastated and betrayed that he had been abandoned by Mike.

After the wedding and just before leaving for Vietnam, Nicky made Mike promise that he would never ever leave without him under any circumstance. And Mike assured him of this. Everybody looked up to and depended on Mike.

After hospital discharge, Mike was able to locate Nicky but lost him. But Mike could have tried harder to track him down after being driven away by the owner of the gambling den who had just recruited him.

Mike was more in a hurry to fly back home to meet Linda (Nicky's fiancee). And he felt guilty and ashamed of this afterwards. Hence the journey back to Vietnam to find him.

I believe this is why Nicky refused to recognise Mike and spat on his face when Mike said he loved him.

To Mike "One shot" meant - Just one more shot then we go home, but to Nicky it meant - His very last shot.

These are my thoughts anyway.