Did George do the right thing by murdering Lennie?
Yes, George felt compassion towards Lennie and he sought to relieve Lennie of all the trouble he is facing. What George had done to Lennie mirrored the incident of what Candy wished he had done to his dog, an act of kindness to relieve suffering. George realized that even though he liked Lennie like his best friend, cared for him; it would be best to murder Lennie to relieve his sufferings. Like euthanasia today, it is comforting to have a person's caregiver nearby in the event of one's death. Today it is referred to as “mercy killing”.
After Curley found out about the death of his wife by Lennie, he organized a search to hunt down Lennie and bring him to justice. At this time, George also realized that there was no place for Lennie to go, and the pain he would suffer and the intimidation at probably the life imprisonment at jail would be horrible. George also came to the shocking revelation that he had no choice but to kill Lennie to avoid him facing the inevitable and crueler future. Also since Lennie wouldn't survive on his own, I think it was best for George to gently kill him than let him face all the torture, pain and suffering he would have to experience in his near future if he did not die as he was a murder criminal.
It is also important to note that George deeply regretted having to kill Lennie, although he understood that he was virtually left with no option and was forced by the circumstances that forced him to take that action. The guilt he feels along with Slim that Curly and Carlson doesn't experience is emphasized in “What's eating those two?”(Steinbeck, 107). These turn of events leave this book with a tragic but memorable ending.