The details of the fine will remain confidential in line with Chelsea club policy but Buck said it was the heaviest "by far" they had ever meted out to a player.
He 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."
He added: "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.
"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."
The former England captain was suspended for four matches and fined £220,000 by the Football Association for racially abusing Anton Ferdinand during a Premier League game between QPR and Chelsea 12 months ago.
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, although the club have fined him.
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.
"What you've got to take into consideration as well is that 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," he said. "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."
Asked if Chelsea should have sacked Terry to send a clear signal that the club takes a zero tolerance stance on 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."
Source: PA
Source: PA