I think the best answer to this question is rather to suggest that the novel does not have a climax at all. Fitzgerald, in his portrait of decadence and dissipation, gradually diminishes Anthony and Gloria's lives until there is little left. Their fall from grace is not glamorous or dramatic like their former lives; it instead shows them dissolving into rootless and pathetic figures.
I would say it climaxes when Anthony snaps, throwing the chair at Dot. His behavior, while slowly growing increasingly erratic throughout the novel, takes a truly desperate plunge near the end as he starts drinking hopelessly, loses his friends, grows violent, etc. Just three pages from the novel's end (446), Fitzgerald narrates, "...then a thick, impenetrable darkness came down upon him and blotted out thought, rage, and madness together-with almost a tangible snapping sound the face of the world changed before his eyes..."
Before this point he had some hope of salvaging scraps of his sanity. After, he is utterly damned.