Gene, as the narrator, actually states what would have made Finny unfit for military service even if Finny had not been injured. When he tries to comfort Finny in the hospital, Gene expresses to his friend that Finny wouldn't have been any good at war because he wanted too badly for everybody to get along. Gene gives the example of Finny wanting to play fun games with the enemy instead of fighting against them. Gene's observation of Finny's being unfit for war is valid. While Finny certainly possesses a charismatic personality and has the ability to get others to follow him, Finny himself usually does not know where he's going or what he's going to do when he gets there--it's all about adventure and camaraderie (two goals which do not have a place in combat).
One of the most important considerations is the incredible innocence and naivete of Finny, qualities that would seem to lead straight to disaster in any kind of combat situation. He also expresses, with some frequency, attitudes that go against the prevailing authority, something that is never accepted in military training or particularly during wartime at the front.
Of course the final straw is that after his accident, Finny is disabled and would be labeled as 4-f, unacceptable for military service due to a physical disability. But this really only masks the real emotional and psychological conditions that made him a very unlikely candidate for military service.