He said to them, "Doubtless you will quote to me this proverb, 'Doctor, cure yourself!”' And you will say, “Do here also in your home town the things that we have heard you did at Capernaum."
I believe he is saying we have the power to heal ourselves if we will use it.
It is not Jesus who says (in the NRSV) "Doctor, cure yourself." He is quoting from an old proverb and attributing it to the Nazarenes who have just heard him speak in the synagogue. He goes on to say, "No prophet is accepted in the prophet's hometown." Too much familiarity with Jesus as a mortal man ("Is not this Joseph's son?") can make it more difficult to hear the truth that comes from Jesus Christ in his elevated consciousness.
In this context, I find it difficult to associate the proverb with the power to heal. Jesus certainly addresses that often in his ministry. I think the old proverb tells us to focus on our own spiritual strength before we presume to judge others. Our ego minds love to diagnosis the failures of others; the Spirit we truly are understands that we will only heal others as we clear the obstacles that block that ability from expressing.