Enlighten me, then. What wars were fought in Europe after 1945?
France fought against Spain? Or UK fought against Germany?
Some wars I can think of after 1945 were the Vietnam War, fought by Americans, and the Iraqi War, fought by Americans.
In fairness, the whole reason Europe stayed peaceful was because it was under control of two major factions who were prepared for war, but neither side wanted to be the first to fire the first shot. However, just because there weren't wars
in Europe didn't mean that Europeans weren't still involved in war to some degree, some of the same wars fought by Americans.
At this point, I don't think there'd be any reason for any NATO member to fight each other.
Vietnam was a hot war which was part of the larger Cold War, which was presented mainly as an ideological war of "the free world" vs. "the evil empire." It's all water under the bridge now, but it seems America's foreign and military policies were geared towards propping up what was left of the global status quo after WW2 - at least what they could salvage of it. They had to make some concessions to the Soviets, because they were once our allies, and there were instances where not every endeavor was successful, such as in Vietnam. There were times when the US government could handle things without even using US troops, such as in coups we supported in places like Iran and Chile.
The wars were fought along the periphery, without going directly head-to-head with the Soviets in Europe - although that was always on the back-burner as a possibility.