France never withdrew from NATO in and of itself; however, it did withdraw from NATO's joint command structure. The history of France's withdrawal from the NATO joint command structure is inexorably linked to that of De Gaulle. u/Aleksx000 's answer to why the Fourth Republic fell (https://www.reddit.com/r/AskHistorians/comments/8cozye/why_did_the_french_4th_republic_collapse/) goes in-depth as to why France felt that her national prestige was diminished post-WWII and post-Suez, and how that led to the Algerian War and the rise of De Gaulle. De Gaulle did not trust the United States, even during wartime, and that feeling was reinforced by the United States siding with the Soviet Union in condemning Britain, France and Israel for their roles in the Suez crisis. His relationship with Britain was also strained, as he had a personal Anglophobia in addition to his suspicion of the Anglo-American "special relationship." He had seen that after the Suez crisis, Britain was reduced from being seen as one of the world's three superpowers to merely being a very junior partner in the Anglo-American alliance. He wanted to avoid France becoming America's lackey, and to that end he tried to promote a third block standing neutral between the US and the Soviet Union. This was his main impetus for reconciliation and rapprochement with Germany, particularly as he had a very close relationship with West German Chancellor Konrad Adenauer (in power 1949-1963). He mistrusted American hegemony over Western Europe and feared the reduction in France's status. This is why France developed an independent nuclear deterrent and left NATO's command structure. The post-Suez wounding of national pride and De Gaulle's mistrust of the Anglophone powers stemming from both geopolitical realities and personal experience from wartime left France wanting to take a path independent from the US and Britain. TL;DR De Gaulle was bitter about not being seen as a superpower post-Suez, and jealous/mistrustful of the US/UK relationship and US power, so he tried to steer his own path independent of NATO and the US.