If your idea of justice is restitution, then it does not matter what motivates a wrongdoer to right his wrongs. There can be various incentives that motivate a wrongdoer to right his wrongs. Off the top of my head I can think of: threat of retribution, growing a conscience, or wanting to gain acceptance to a community (shunning can be a powerful motivator). We do not live in a world where all wrongdoers grow a conscience and right their wrongs. Most wrongdoers do not right their wrongs even if they grow a conscience. So sure, violent retribution is not needed in order for their to be restitution. But for their to be law there must be at least the threat of violence. So we can imagine hypothetical worlds where people do no wrong to others, or if they were that somehow they right their wrongs from their own self-motivation, but this is just an imaginary world. In the real world, law is the primary mechanism people use to right wrongs, and law requires at the very least the threat of violence.
I recommend reading A Praxeological Account of Law by Clayton.