We may not be healing each other, but Jesus does. In fact, it's what He most often gets in trouble for in the Gospels - healing persons (which kind of makes you think maybe we're onto something; no healing, no trouble). It's really not that Jesus is healing persons that gets Him in trouble. Even the Pharisees can't reasonably claim they'd rather the blind man stay blind. No, what Jesus gets in trouble for is healing persons on the Sabbath.
But He kind of has to.
He has to heal persons on the Sabbath because that's the only day that they're not hard at work trying to heal themselves. And if you want to make sure God gets the credit for doing something absolutely amazing, then you have to make sure that there's no way that persons can take the glory for it.
Think about it - for six days a week, the afflicted are doing everything they can to either manage their affliction or cure it. Blind men are going around trying to make sure they've got all the supports that they need to make it through another week as blind men. The bleeding woman, we're told, exhausted all of her resources trying to find doctors who could make her well. You can bet that the man with the shriveled arm spent a lot of his time trying to figure out what to do about it and that the woman hunched over spent a lot of her time trying to stand up.
All of that stops on the Sabbath. Every bit of it. There's nothing to do for yourself on the Sabbath because there's no one to help you. Doctors aren't working. Pharmacies are closed. The markets are shut down. Persons are not walking through the streets, lest they mistakenly be taken for doing some kind of work. If you need something on the Sabbath, too bad. There's no one to help you, and you certainly can't help yourself.
That's why when we see Jesus healing someone, we see Him healing them on the Sabbath. On the day of rest, there's no mistaking Who is really at work.
Funny, isn't it?
In the beginning, God created the heavens and the earth. On the seventh day, He rested. And then Israel instituted the practice of the Sabbath to remember God's creative rest and to honor it. Even God doesn't work on the Sabbath. He's set the world in motion so that He doesn't have to.
Fast forward a few thousand years, and God's back at work on the Sabbath. It's the only day He can know for sure that He gets the glory.
No longer is it that creation has been established to work for itself, to stay in motion, to keep going even while God rests. No. Now, God needs creatures to rest so that He can remind them of who He is. He is the One who makes things happen when there's absolutely nothing you could do for yourself. When there's nothing you can do but hold your breath and wait for the next chance to go back at it, God breathes His life back into you, and you're healed.
Pretty cool, huh?