Terry was sacked as England captain by Capello earlier this month, but in a statement released by his lawyers, Bridge said his position in the national squad had become "untenable and potentially divisive".
England face Egypt in a friendly at Wembley next Wednesday and despite reports earlier this week suggesting Bridge was considering his international future because he was unwilling to play alongside Terry, Capello believed he would be available for selection.
The Italian stated on Wednesday: "No, he will be with us. No problems about the other things."
However, Bridge's statement has shown the left-back remains troubled by the situation.
"It has always been an honour to play for England. However, after careful thought I believe my position in the squad is now untenable and potentially divisive," said Bridge.
"Sadly therefore, I feel for the sake of the team and in order to avoid what will be inevitable distractions, I have decided not to put myself forward for selection."
With England's opening World Cup match against the United States on 12 June less than four months away, Bridge's decision leaves Capello with a selection headache over England's left-back spot.
Bridge decision terrible for Capello - Farquhar
Chelsea's Ashley Cole is already sidelined with a broken ankle and is a doubt for the tournament.
Aston Villa defenders Stephen Warnock and Nicky Shorey, Everton's Leighton Baines and Manchester City's Joleon Lescott are all possible replacements, although all are relatively inexperienced at international level.
Bridge, who has played for England 36 times and recently returned from a two-month lay-off with a knee injury, was a team-mate of Terry's at Chelsea before joining Manchester City last year.
The pair are due to meet on the pitch for the first time since details of the alleged affair were made public when City visit Chelsea on Saturday.
Only a fortnight ago, City boss Roberto Mancini expressed confidence that Bridge would go to South Africa.
He admitted on Friday: "I thought Wayne would stay with the national team.
"It must have been a very difficult decision because I know how important the national team is for any player."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.