The 31-year-old had until 6pm today to make a decision on whether to fight the suspension but it was expected that the centre-back would take the forced absence.
Terry released a statement, which said: "After careful consideration, I have decided not to appeal against the FA judgment. I want to take this opportunity to apologise to everyone for the language I used in the game against Queens Park Rangers last October.
"Although I'm disappointed with the FA judgment, I accept that the language I used, regardless of the context, is not acceptable on the football field or indeed in any walk of life.
"As I stated in the criminal case, with the benefit of hindsight my language was clearly not an appropriate reaction to the situation for someone in my position. My response was below the level expected by Chelsea Football Club, and by me, and it will not happen again. Looking forward, I will continue to do my part in assisting the club to remove all types of discriminatory behaviour from football."
The former England international will miss Chelsea's trip to Tottenham Hotspur this weekend as well as a league and League Cup fixture against Manchester United and an away match against Swansea City.