Terry was reported to have been left distraught after being told of Capello's decision during the 12-minute meeting at the FA headquarters at Wembley stadium.
"I fully respect Fabio Capello's decision. I will continue to give everything for England," Terry said in a statement.
The Italian coach retained Terry as skipper when he took the England job in 2008 after the sacking of Steve McClaren, the then-manager, and since then the team have flourished, qualifying easily for the World Cup in South Africa in June.
However, the suitability of the married father-of-two for the role has been repeatedly questioned in the media since the story emerged following a high court decision to overturn a ban on reporting details of the alleged affair.
Capello said that Manchester United defender and current England vice-captain Rio Ferdinand would take over the role.
"When I chose John Terry as captain, I also selected a vice-captain and also named a third choice. There is no reason to change this decision," he said.
However, Ferdinand has featured only spordically for United this season as he has struggled with a number of injuries, putting his participation in this summer's World Cup in doubt.
Liverpool midfielder Steven Gerrard is the current third choice for the role, but Capello has also spoken of the leadership qualities of Wayne Rooney, the Manchester United striker who led England in a friendly against Brazil in Qatar.
Chelsea coach Carlo Ancelotti has insisted that Terry's private life will have no bearing on his position as the club's skipper.