The French/American realtionship is far too complicated to boil down to just an Iraq or military support thing. There are economic, political, and cultural tensions at work.
France is currently at work trying to solidify its position within the EU. The EU is being formed to combat American economic dominance and the potential dominance of behemoths like China.
Culturally, the French make no secret of their resentment towards the invasion of American culture and slang into their society.
Politically, there are many power struggles within the UN and internationally over America's use of the death penalty, import duties, trade relationships with other countries, etc.
For those who say the French are all a bunch of cowards, it may be a fun joke but their military strength throughout history and their military technological prowess is among the greatest in the history of the world. The French foreign legion, the French resistance during WWII, and the French military forces fighting in places like Afghanistan and in UN peace keeping forces are highly trained and very professional.
There are significant reasons for the tensions between France and the US and they are often stoked by very public refusals of US requests on the world stage (not the least of which was allowing planes to fly over French air space when we bombed Libya). The mistaken bombing of the French embassy by US planes in those raids didn't help their impression of the US either.
There're lots of reasons for the dislike, Iraq is but one part.
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