The Chelsea defender was suspended by the Football Association for abusing Queens Park Rangers defender Anton Ferdinand in a heated west London derby last October.
The incident revived the issue of racism as well as bringing shame on Terry and Chelsea, which said it will take disciplinary action against the 31-year-old centre back.
"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," Terry said in a statement.
Terry has always claimed he only repeated the obscene phrase he was accused of saying by Ferdinand.
Terry was cleared in a criminal trial in July of a racially aggravated public order offence but was found guilty by the FA on a lesser burden of proof, with the governing body's disciplinary panel finding his defence "improbable, implausible and contrived."
The ban was half the length of that given to Liverpool striker Luis Suarez for racially abusing Manchester United defender Patrice Evra during another Premier League match last October, while Terry was also fined a record 220,000 pounds ($356,000).
Having retired from England duty in protest at the FA's pursuit of the high-profile case and having lost the national team captaincy, Terry has had the recent two-week international break to discuss his next step with lawyers.
"After careful consideration, I have decided not to appeal against the FA judgment," Terry said. "My response was below the level expected by Chelsea Football Club, and by me, and it will not happen again."
Chelsea wouldn't reveal what sanctions the club had taken against Terry, the captain of the European champions, saying the "confidential" disciplinary action is "in accordance with our long-standing policy."
"Chelsea Football Club believes John Terry has made the correct decision by not appealing against the FA judgment relating to language he used at the QPR match last October," a club statement read. "Chelsea also appreciates and supports John's full apology for the language he used. The club firmly believes such language is not acceptable and fell below the standards expected of John as a Chelsea player."
His suspension will start immediately, meaning he misses Premier League games against Tottenham, Manchester United and Swansea, as well as a League Cup fourth-round match against United.
If Terry is wearing the captain's armband when he returns to action — he will next be available domestically for Chelsea's league match against Liverpool on Nov. 11 — then the club is likely to have only punished him with a fine.
Chelsea, which has stood by Terry throughout the feud, has a policy of banning fans for life if they are found guilty of racial abuse.
"Looking forward, I will continue to do my part in assisting the club to remove all types of discriminatory behaviour from football," Terry said. "I am extremely grateful for the consistent support of Chelsea FC, the fans and my family."