What's the difference between fight and battle when they are used as a verb?
Solution 1:
Two points.
Firstly no matter how big the battles were in which your grandfather took part his individual contribution would have been on a personal scale.
Secondly the description of doctors 'battling' to save a child's life is a metaphor. They are not fighting like soldiers they are doing their very best, medically, to save the child. They aren't fighting against other people, they are working against the infection or other physical condition that threatens the child.