Legal Question in Civil Litigation in California

Generally criminal statutes have an implied right of action. Some set damages. This is complicated to explain unless more research is done just a general answer. So you could sue for that tort based upon a criminal act but could possibly sue for battery, negligence, negligent infliction of emotional distress, etc.

Usually criminal courts though grant restitution.

You absolutely can sue in civil court. Damages would include costs of removeal and reconstructive plastic surgery, pain and suffering, probably others, but it would depend on the details of the situation. Any good general litigation attorney would handle a case like that.