David Fincher's Gone Girl won 0 Academy Awards.
Gone Girl received one Oscar nomination at the 87th Academy Awards, held in 2015: Rosamund Pike was nominated for Best Actress for her performance as Amy Dunne. She did not win, and the film received no other Academy Award nominations, so its total was zero wins.
The nomination recognized Pike's controlled, unsettling performance in Fincher's adaptation of Gillian Flynn's novel. The film also attracted major awards attention for Fincher's direction, Flynn's screenplay, and Trent Reznor and Atticus Ross's score at other ceremonies, but those categories did not translate into Oscar nominations for Gone Girl. The Academy's recognition was limited to Pike's acting performance.
This question can be confusing because Gone Girl was widely praised and earned many nominations from critics' groups and organizations including the Golden Globes, BAFTAs, and Screen Actors Guild Awards. It also won awards outside the Oscars, including honors for Pike, Flynn's adapted screenplay, editing, and the film's thriller production. Those wins are sometimes mistakenly counted as Academy Awards. The correct distinction is that Gone Girl was an Oscar nominee, not an Oscar winner.