ch. 6 in the novel
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George isn't justified, but he wants to do it himself and quickly before Curly or anyone else does it hatefully with a rope around Lennie's neck.
George has no choice. If the mob didn't torture and kill Lennie, they would (at the very least) lock him in a cage. As Slim said, "that ain't no way to live". George could not let the situation go any further, he could no longer protect Lennie.