Davis, 23, suffered the injury during a training camp for the Rio Games and has been replaced by 24-year-old Ruaridh McConnochie in the 12-man squad.
Coach Simon Amor said it has been a "tough year" for Davis and described the setback as "incredibly disappointing".
Team GB make their first appearance on 9 August against Kenya.
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Alex Davis has been ruled out of Great Britain's rugby sevens squad for the Olympics because of an ankle injury.