The question is why would Satan want to heal the sick?
1. To deceive
But how could his deception profit him if the person performing the miracle is doing it in the name of Jesus. Many who are accused of performing miracles through Satan usually call upon the name of Jesus in order to perform the miracle. Wouldn't Jesus then get the credit?
It is believed that disease, sickness, and infirmities are brought about by demonic spirits as stated in the bible. Jesus healed some with infirmities by casting out demons. so if Satan/Devil were to go around casting out his demons how long would his kingdom last, and what threat would he be and to whom other than himself?
It would almost be like the expression of selling guns to the Indians. Jesus said any kingdom which operated like this could not stand.
The only way Satan could benefit from healing the sick is if he could prove that the miracle did not come from God. Therefore they are some who try to deny the power of God. The magicians in Egypt in the bible were trying to match miracles performed by God to convince Pharaoh that the miracles done Moses weren't of God, but were soon over taken by the awesome power of God.
So can Satan heal the sick? The answer is no, because for Satan to have persons cast out his demons in the name of Jesus would be shooting himself in his own foot to the credit of God Almighty.