TLDR
- Moved to answer for potential spoilers...
- Of course they do; we just don't know what...
They do, but I don't think we know much about it. If we knew what they did to heal we would likely consider it to be magic of some sort. There are two things we know for sure about.
They use a "Soul Forge""Soul Forge" to diagnose illnesses and that somehow the "Odin Sleep" kept Odin alive for a little bit longer than is typical.
Beyond this I do not think we know much. I can not remember any points in the MCU or comics where an Asgardian pulled out a med-kit or anything. However from Ragnarok we do know:
Thor and Loki used to play "Get Help" where Thor would be dragging Loki (and then he may have thrown Loki at them?!) likely to distract guards or something when they were younger - From this we know that Asgardians do help the wounded and that there are procedures in place to help those that are hurt.
Towards the afterlife, we know:
Thor also said a prayer for his father after he passed stating that he would go to Valhalla - This tells us that they believe in an afterlife and that (if it follows typical Norse mytho) dying on the battlefield is the best way to die. Saving someone from that fate could be seen as an insult? Maybe they allow the warriors to fall out of respect.
