The former England captain was suspended for four matches -- starting with Saturday's league match against Tottenham -- 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 confirmed he would and that he would not be given an additional ban by Chelsea, although the club have fined him.
"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 said on talkSPORT radio.
Details of the fine would remain confidential in line with club policy, Buck said, adding that it was the heaviest fine "by far" they had ever meted out to a player.
"We have kept in 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," Buck said.
"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."
Buck also apologised to the Ferdinand family for what they have been through, something Terry, controversially, did not do in his statement on Thursday.
"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."
Chief executive Ron Gourlay added that there was no question of Chelsea sacking Terry to send a clear signal that the club has a zero tolerance to racism.
"We haven't tolerated the language that John Terry used. We've said it was not acceptable and we have taken action," Gourlay said.
"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."
Source: AFP
Source: AFP