In the view of Augustine and the Judaeo-Christian ethic in general, the taking of an innocent life is immoral. And even given the most charitable interpretation of George's shooting of Lennie as a "mercy killing," from our perspective today, it would appear unjustified to most people.
However, in the context of the time when Steinbeck's novel takes place, George does not have the options that are normally available to us today. Rural California in the early twentieth century, like much of the country in general, was a rough place. Migrant workers such as George and Lennie did not have the protection of the law against the farm owners. It's obvious to George that Lennie is going to be lynched by Curly and the other farm hands for having killed Curly's wife, in spite of the fact that this was an accident. Even if they had captured him and turned him over to the police, Lennie would probably have suffered cruel treatment in jail, given that there was relatively little sympathy for developmentally disabled people at that time. Under the legal system, he would have been convicted of murder and executed. George thus takes matters into his own hands and gives Lennie what he believes is the least cruel solution: an instant, painless death. Steinbeck's underlying theme, as in The Grapes of Wrath, is that of the unfairness of the whole system toward the rural laboring class. Moreover, the manner in which George must conceal from the others on the farm that Lennie is developmentally disabled is an indication of how unfair attitudes were toward disabled people. Economic, class, and societal pressures converge to create a situation in which George feels that killing Lennie, as bad as this is in itself, is the only option.