"To heal" means to make healthy. It's more abstract. The procedure of fixing damage to a limb essentially repairs it. It's just an organic machine, and as such any procedure to fix a problem with it is a repair.
For instance, if Alico is infected with influenza, then an antibiotic will heal Alico, and the system will repair any damage caused to itself by the virus. Damage gets repaired; health gets healed.
I'd go so far as to say that the most common form I've heard has been "fix", and that is a reasonable translation as it would be synonymous to "repair" in this context. I've honestly never heard the phrase "heal the leg" before regarding invasive procedures. In fact, I'd say I've heard more frequently "the leg heals" as an intransitive verb, which does not mean the same thing (we're talking about making a repair to a limb in the original sentence).