The FA wanted to enter teams for the tournament, but had to scrap the proposals after encountering opposition from the football associations of Scotland, Wales and Northern Ireland.
“After careful discussion, The FA has decided not to enter either a women’s or a men’s team into the Rio Olympics 2016,” the FA said in a statement.
“We are disappointed not to be able to go ahead, given the fantastic opportunity it would have afforded the players and the broader exposure it would have brought to the game in our countries.”
Great Britain fielded men’s and women’s teams at the London 2012 Olympics, where both sides were eliminated in the quarter-finals.
But Scotland, Wales and Northern Ireland are unwilling to repeat the experiment as they fear jeopardising their status as independent countries in the eyes of world governing body FIFA.
Trefor Lloyd Hughes, president of the Football Association of Wales, said that he had been “livid” when the FA informed him of its plans in a letter in January.
FIFA vice-president Jim Boyce had previously stated that he had been given “an absolute categorical reassurance” from FIFA that British teams would only be able to compete at the Olympics if all four home nations agreed.