Animals naturally do regenerate health, albeit at a very slow rate. I don’t believe you can do anything to speed up the process, you can only try to prevent them from getting in any more fights and thus taking more damage.
I’m also pretty sure that tame animals won’t fight each other in the base game. If you have PK’s rebalancing on or Ascension they might because I believe PK’s does something about how monsters decide to aggro on each other.