The former England captain was suspended for four matches and fined £220,000 by the Football Association earlier this month for racially abusing QPR defender Anton Ferdinand.
Terry decided this week not to appeal the decision, leaving Chelsea to decide what action of their own to take.
Manager Roberto Di Matteo refused to say on Friday whether Terry would remain captain but Buck has now confirmed he would and that he would not be given an additional ban by Chelsea, although the club have fined him.
Buck said on talkSPORT: "John will continue to be captain of the club. We have taken disciplinary action and we think it is firm disciplinary action and appropriate for the circumstances."
Buck insisted the details of the fine would remain confidential in line with club policy but said it was the heaviest fine "by far" they had ever meted out to a player.
He said: "We have kept it mind first of all that a court of criminal law found him not guilty. Of course we also note the FA decision and we certainly respect that decision. We've kept these factors in mind, we've consulted with the owner Mr Abramovich.
"This is not a decision we've taken lightly, it's not a decision we took in a half-hour meeting, we discussed it over a long period of time and we think we've taken appropriate disciplinary action."
Chief executive Ron Gourlay argued Terry's long service to Chelsea should also be taken into account but condemned his behaviour in the Ferdinand incident.
He said on talkSPORT: "What you've got to take into consideration as well is John's played over 550 times for the football club, he's captained the side over 400 times, he's led the club and the team tremendously well during these games.
"We believe this was an error of judgment, it was out of character for John. He did fall below the high standards we expect at the club. The language used on the day, whatever the context, was wrong. We've come down on John very, very heavily.
"We are not sweeping this under the carpet. We have had to deal with this for the last 12 months. John let himself down, he let the club down, the words were inappropriate that he used."
Buck insisted the issue would not be forgotten once Terry returns to action, saying: "We would like to draw a line under this but, by drawing a line under it, it does not mean we want to forget it.
"We know the Ferdinand family will remember it, we know John Terry will remember it, we know the media will remember it and we will remember it too."
Asked if Chelsea should have sacked Terry to send a clear signal that the club has a zero tolerance to racism, Gourlay said: "We haven't tolerated the language that John Terry used. We've said it was not acceptable and we have taken action.
"We don't believe for one minute that John is a racist and we must not forget that he was cleared in a court of law. We have taken action and we hope it gives out a clear signal that it is not acceptable.
"We know that not everybody is going to agree with this decision but we have tried to focus on improving and on doing better. The club is disappointed, the player is disappointed we are all disappointed and we have got to put things right."
Buck added: "I think we can say that with John Terry at the club and captaining the club we can move forward from this incident but we won't forget this incident.
"We think we have done what is appropriate. You can either agree with that or not."
Buck also addressed the punishment meted out to Ashley Cole following his foul-mouthed tweet directed at the FA after his evidence was called into question in their report.
The defender later apologised in person to FA chairman David Bernstein but was charged and fined £90,000.
Buck questioned the sanction, saying on talkSPORT: "The club was surprised by the level of the fine imposed on Ashley Cole.
"I don't want to sound like an apologist for him. But he apologised to David Bernstein and the England manager and we would have viewed that as mitigating circumstances."
Ferdinand has refused to shake hands with Terry or Cole since the incident and has himself come in for abuse from the stands at Stamford Bridge and other grounds.
Buck apologised to the Ferdinand family for what they have been through, something Terry did not do in his statement two days ago.
Buck said: "I understand, and as a club we understand, what they have gone through since a year ago and Chelsea Football Club would like to apologise to them. It has been a very difficult year for them and they didn't do anything wrong. And we are sorry."