Didn't he stand a better chance of fighting back, given he has a gun, and was good with it? Instead, he wasted the little energy he had left so he can sit at a river...
to enjoy his last moments.
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In reality people often do strange or unusual things when they suffer a traumatic, and in Mike's case, a mortal injury. Their mind continues to work as it does and they realize that they need to DO something. However, after a short period of time, their injury overcomes their adrenaline rush and they are forced to halt whatever it is that they are doing.
In Mike's case, being a trained professional, he would have tried to flee since he knew that he was wounded and that he was unable to defend himself. It seemed though that his body wouldn't allow him to go much further than it did and he simply sat down a short distance from the car where he'd been shot. When this occurred, he probably realized that his wound was a mortal one and he simply resigned himself to his fate.
There's also the reality which may have occurred to Mike that there was no place for him to go. Even if Walt had called an ambulance and they arrived before he bled to death, he would have been arrested and gone on trial for at least the distribution of the money to his criminal conspirators if not been tried along with them for the events which occurred during the Fring drug organization. Sitting by a river and dying there may have been a far more pleasant option for him than spending the remainder of his adult life in prison.
I don't think its that simple, everything within BB is chosen with almost OCD-like meticulousness. Maybe it's an a symbol of Mike's psyche? I mean, that, in terms of geography, it could mean this. Let me explain/speculate.
Walt is often associated with desert imagery, and this is because Walt is a isolated character. He's the desert because literally no life can survive there without having to go to extreme measures. Even the title to the last season was a reference to the desert. "remember my name" it promoted, whilst earlier in the film we hear"....in the desert you can't remember your name, cause there ain't no one who can give you no pain". Walt constantly echoes (through his singing of America's song) this theme of isolation and furthermore unexpected death.
So I would hope to associate Mike to a river, or maybe regret to the river (As it was a river where Hank literally threw his one opportunity to live away). As it flows away, maybe he feels a sense of regret for an innumerable amount of reasons.
In a word, the answer is what you've said
to enjoy his last moment.
But let me elaborate, if you watch Mike carefully, he's an enforcer. He had only two things in his life that mattered to him: Gus Fring and his granddaughter. For some obscure reason Mike is loyal to Gus which was not shown in the show. After Gus's death, his granddaughter is the only thing in his life. Because of her, Mike agreed to become a partner of Walt even knowing that Walt is a time-bomb. After the arrest of his lawyer and confiscation of his share. It' literally impossible for him to see his granddaughter for not only the Police, he might not have the gut to face her.
So after being shot, his life is literally over. What options did he have? Without immediate medical attention, he won't survive. An immediate medical attention isn't possible without involving Police.
And why he didn't try to kill Walt? He had already enough blood on his hand. At his last moment, he just didn't want more blood.