Is it important to know that the narrator of the story an American soldier?
I agree with the second post here in saying that the importance of the narrator being American really has to do with the setting of the story more than the narrator's "American-ness". Using a non-Italian narrator is essential to the meaning of the story as it seeks to express ideas of isolation, other-ness, and despair. The narrator's nationality is yet another challenge that faces his sense of self, sense of potency, and sense of the nature of his future.
It is important to understand that the narrator in "In Another Country" is an American soldier because one of the themes of the story is isolation. The fact that the narrator is an American in a foreign country isolates him. Most of his companions are not Americans, plus he has been wounded and is, therefore, isolated from the battlefield. Even the setting of the hospital is isolated because is it "beyond a bridge" separated from the main part of the town.