Technically, of course, the person who pulls the trigger and kills Lennie, is George.
However, it is also arguable that Lennie is responsible for his own demise due to his total incapability to control the triggers in his environment. This incapability led him to commit the ultimate mistake (although not on purpose) of taking the life of Curley's wife. The fact that he gets to that point is quite dangerous, making it understandable (though not entirely justifiable) that George just decided to mercy-kill Lennie, rather than have him get caught by the lynch mob that was coming Lennie's way.
Lennie is, so to speak, a not-so-gentle giant with immense strength. The problem is that he operates purely out of impulse. As such, he destroys many things in his way, although not on purpose. Realistically, he is out of control. George can only do so much to curb Lennie's actions.
Here is the evidence why Lennie may be held responsible for his own demise, even though he does not do things "on purpose". From the very start of the novella, we see foreshadowing. George complains about all the things that Lennie has been responsible for. All of those things are negative and caused by Lennie's inability to comprehend how strong he is, how emotional he gets, and how physically controlling he can be.
“God, you’re a lot of trouble,” said George. “I could get along so easy and so nice if I didn’t have you on my tail. I could live so easy and maybe have a girl.”
From George's own mouth we learn that Lennie:
- "Petted" a woman's dress the way he would one of his mice, causing for her to yell for help, and driving both George and Lennie to hide and escape the county.
- Has killed animals by accident just because he is too rough with them and cannot discern was is a "good touch."
- Got George in trouble several times throughout their lives precisely because of his incapacity to respect boundaries and keep his hands to himself.
Essentially, we could bring it down to social Darwinism: survival of the fittest.
In Of Mice and Men, we see that those who "make it" in the ranch are the farm hands who have what it takes to survive that harsh, dry, isolating, and cruel environment. This is evident in Slim "the prince of the ranch"; Crooks, whose sour demeanor and lonely lifestyle are factors that help him survive there; and George himself, who holds himself together even with Lennie by his side.
In all, Lennie is just too impulsive, too unprepared, and too overwhelming to survive in such an environment. Sooner or later, he would (as per his history) have done something that would entirely mess things up. Killing Curley's wife was the ultimate mistake, which would have brought with it the ultimate punishment. George knew what was coming, so he just pulled the trigger and killed Lennie in one shot rather than have him go through the harshness of a lynch mob.