If you have ever been near witch doctors in tribal situations, etc., you wouldn't say so simply that Satan cannot heal. There is no doubt that witch doctors have power that heals people (or kills people for that matter).
Now of course I agree that God is all-powerful and alone is the one who can give life and who alone is the Healer.
But when man sins and is controlled by sin, then God allows Satan access and power over people's lives. I wouldn't want to be healed by Satan because it gives him power and control over your life. I have heard of people who have had to rebuke "healings" when they were born again.
I would say that only God is the Healer, but that Satan can "heal" in situations where man by sin allows him to enter.