I’m currently reading through “Armies of Heaven” by Jay Rubenstein and he’s now mentioned a few times where ladders would break during a siege when too many soldiers attempted to climb them at the same time. He describes how they fell far to the ground in full armor. No doubt this would probably have led to many men breaking various bones or even to death. This is obviously just one of thousands of ways men were injured in such battles but the ladder example particularly made me wonder what would happen to a soldier who broke their legs in such an event. The book doesn’t talk about logistics much besides mentioning when they were low on food or material to construct siege engines so my question is, what would become of these soldiers with them being so far from home and with the crusader army often being short on resources like at the siege of Antioch? Did they have the capacity and ability to treat such injuries or would they send conveys of the injured to the closest controlled city or was it like a put them out of their misery kind of thing for severe injuries? Thanks in advance.
In terms of injuries, even small ones that would seem insignificant today could result in infection. That's a very important part to remember when talking about any time in history before antibiotics came into use. A simple gash, cut, splinter or scrape could result in infection and gangrene if not carefully taken care of with clean bandages, clean water, or some kind of antiseptic salve.
Additionally, in the case of broken limbs, some minor ones would be stabilized With a splint, while the major ones would be amputated. Again, infection could be problematic in this case. Provided that the limb wasn't amputated, there was no guarantee that the person would return to former function. Even with today's medicine bones can heal incorrectly, resulting in complex surgery. Back then, however, a person - especially a poor soldier - would just have to live with a limp, or a poorly functioning limb.
In terms of complex or severe injuries, you're right, the person would be given their last rights - if possible - and if they were going to die, "it is God's will". While dying, they may be given something for sleep or pain, maybe even alcohol as an alternative to medicinal concoctions. Whether they were put out of their misery, I can't say. Perhaps a case could be made for doing it on the battlefield, but I'm sure killing your own men wouldn't bode well, plus killing your own side might be seen as a sin. I would more likely assume that if they were at a medical area, they would give them enough pain and sleep medication until they passed, at which point they would be buried properly - if the setting allowed.
Another thing to note is that falling from height could easily kill someone, especially in heavy armour. Spinal chord injuries, which are difficult enough to treat today would also likely prove to end a life.
Some other factors: Fighting in formation was also very hot, and could be cramped in some instances. There are many accounts of soldiers from Roman times, through the middle ages getting heat stroke, or being severely dehydrated. In formation fighting, if a man fell, he could be trampled to death by either side of the battle.
This link provides some interesting information furthering the topics in this post: https://www.medicalnewstoday.com/articles/323533.php
This link looks at the diversified treatment based on country and time: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6139913/