Just to add a little info on the military aspect:
In the legion, every castrum (fort) had something called a valetudinarium( meaning house of the surgeon; hospistal). This was where all the wounded were brought immediately after battle.
In another aspect, there was a longstanding tradition since the Republic that every general have his own real qualified and top-of-the-line physician who stayed with him everywhere he went, even into battle.
But here's the most important thing about legionary medicus. For common legionaries, the medicus who cared for them were actually their own fellow soldiers. In the legion, you were considered a medicus if were good at attending and healing the wounded soldiers, not much experience needed. It was these medicus whom the common soldiers valued so much and with lots of compassion. For example, when a medicus died, often his comrades of the unit( cohort etc...) would pay respects to him by contributing him a monument with inscriptions of his role in the field. But, the job of medicus to the legionaire himself was a second priority, the job of being a ruthless soldier was the main one. By the time of the Gallic campaigns, Caesar recognized the importance of medicus, thus he created a special staff of them, but rules still applied.
In the legion there were three specified types of medicus:
medicus ordinarius--> obviously he tended his own unit
medicus cohors---> he attended the cohort
medicus legionis (not sure which case)---> attended everyone.