George is ultimately responsible for Lennie's death because George is the one who kills him. This is clearly an unfair statement and therefore, the term "responsible" is just not appropriate to describe George's role here. For George, this was a mercy killing. He'd rather kill Lennie himself, quickly and painlessly, than let Curley or legal officials get to him first. So, George did kill Lennie but he is not responsible for the situations which led him to feel he had to kill Lennie.
Those situations can not be blamed on any one person. Curley's wife was being friendly, albeit too flirtatious and asking for retribution from Curley. But there's really no one person who is responsible for Lennie's death. Lennie, unfortunately, is mentally disabled. In his life, he's only had Aunt Clara and George to look out for him. He lives at a time when jobs are difficult to come by and he finds himself (with George) as an itinerant ranch hand, a difficult lifestyle, working with people, some of whom do not have the education and consideration to deal with Lennie's state of mind.
So, Lennie's death is a tragedy of circumstance. His death is caused by many factors: his mental capacity, his occupation, the people he is surrounded by. Slim is the only one (other than George) who recognized that Lennie's death had to come sooner or later because of these circumstances.
Slim came directly to George and sat down beside him, sat very close to him. "Never you mind," said Slim. "A guy got to sometimes."
If anything, George prolonged Lennie's life for as long as he could.
In his own words, John Steinbeck wrote,
"Lennie was ... to represent ... the inarticulate and powerful yearning of all men."
The killing of Lennie is not simply the shooting by George--it represents far more than the death of a man. For, Lennie has been the keeper of the dream, by George's own admission--he tells the others that in reciting repeatedly the dream of having a farm, he himself began to believe it. Truly, the childlike-Lennie has given George and the others hope.
With Lennie's death, the dream dies and hope is lost. Therefore, the death of Lennie is a metaphor for the despair and desperation of the disenfranchised of the Great Depression--an expression of Steinbeck's discontent with the hopeless social situation of the lower classes in America. So, in Steinbeck's view, America's government is to blame for the death of Lennie.