The Board of Control for Cricket in India (BCCI) has told its players to stay in Sydney rather than travel to Canberra for a tour match on Thursday.
Harbhajan was banned for making a racist remark during their tempestuous defeat to Australia in the second Test.
He was found guilty of breaching the players' code of conduct.
The International Cricket Council held a four-hour hearing after the Test finished on Sunday, finally announcing their verdict deep into the night in Australia.
Australia's players had claimed that Harbhajan called Australia's Andrew Symonds a "monkey" during an on-field incident.
Match referee Mike Procter said he was satisfied Harbhajan had used the word - though neither of the two umpires heard the remarks - and that "he meant it to offend on the basis of Symonds' race or ethnic origin".
All-rounder Symonds, 32, is the only non-white player in the Australian side.
The BCCI's statement said: "The Indian Board realises the game of cricket is paramount but so too is the honour of the Indian team and for that matter every Indian.
"To vindicate its position, the board will fight the blatantly false and unfair slur on an Indian player."
BCCI Stands With Harbhajan As Protests Continue
Cricket Australia chief executive James Sutherland, who is expecting the tour to continue as scheduled, has proposed a peace meeting between captains Ricky Ponting and Anil Kumble to try to resolve the escalating crisis.
We have the threat of a tour being abandoned and once again the ICC has been plunged into a crisis
BBC Sport's Oliver Brett
And BCCI vice-president Rajiv Shukla said a decision on when the players would leave Sydney and resume training would be made at a meeting on Tuesday at 0700 local time (2000 GMT Monday).
Earlier, India team manager Chetan Chauhan said he believed Harbhajan had been harshly treated.
"I told the match referee this is wrong. There was no conclusive evidence from the Aussie side," he said.
And BCCI spokesman Ratnakar Shetty told BBC Radio 5 Live: "Harbhajan says he did not say what has been alleged.
"The match referee has made his case based on hearsay. I don't understand how he arrived at his conclusion.
"Sachin Tendulkar also refused to accept that [Harbhajan] said that word."
Harbhajan was accompanied at the hearing by Chauhan, assistant manager Dr MV Sridhar, captain Anil Kumble and Tendulkar.
If he was to lose the appeal he would miss the two remaining Tests in Australia, and one further match.
Australians are no angels, but that's no excuse for what happened
The 27-year-old has played for India since 1998 and has taken a combined total of almost 450 international wickets in all forms of the game.
He was batting with Tendulkar when the incident took place during Friday's play at the Sydney Cricket Ground.
It prompted Australian captain Ricky Ponting to complain to on-field umpires Mark Benson and Steve Bucknor, who subsequently laid a charge under section 3.3 of the code of conduct.
The section refers to the use of "language or gestures that offends, insults, humiliates, intimidates, threatens, disparages or vilifies another person on the basis of that person's race, religion, gender, colour, descent, or national or ethic origin".
The ICC says it has a zero-tolerance policy towards racism.
Former England captain Tony Greig said a bitter rivalry between Australia and India had developed over recent series, and that players had been warned to be careful about what they said.
Greig told the BBC: "Australians are no angels - you only have to speak to some Indians and some Sri Lankans to know they have dished out plenty themselves.
"That's why India are saying: 'Who are the Australians to talk on this issue?'
"But that's no excuse for what happened and everyone's on notice. This has grown over a period of time."
The third Test between Australia and India is due to begin in Perth on 16 January.
India have reacted to Harbhajan's ban by lodging their own complaint against Australia's Brad Hogg for allegedly making an abusive comment to one of their players, according to some reports.
Harbhajan is the first player punished for a racist remark since South Africa's Herschelle Gibbs was suspended for three matches in January 2007.
He was caught on a stump microphone talking about Pakistan fans.
In 2003, Australia's Darren Lehmann was suspended for five one-day matches after he was found guilty of shouting a "racially-motivated obscenity" in a match against Sri Lanka.
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