The programme, aired on December 28, was criticised for playing on various stereotypes about India, as Jeremy Clarkson, James May and Richard Hammond joked about food, illness and trains.
The High Commission of India (HCI) has now written to the BBC after it was contacted by "too many people" who were "very upset" by the programme.
The BBC, which has received 188 complaints about the Top Gear special, said that it would respond directly to the HCI soon.
A BBC spokeswoman said: "We have received a letter from the Indian High Commission and will respond to them in due course."
In its letter to BBC director general Mark Thompson, quoted in The Daily Telegraph, the HCI hit out at the "tasteless" programme, and accused the BBC of reneging on its agreement.
"The programme was replete with cheap jibes, tasteless humour and lacked cultural sensitivity that we expect from the BBC," the HCI told Thompson.
"I write this to convey our deep disappointment over the documentary for its content and the tone of the presentation. You are clearly in breach of the agreement that you had entered into, completely negating our constructive and proactive facilitation."
Diplomats at the consulate told the paper that they had received "hundreds" of letters and phone calls from people upset over the programme.
One diplomat said: "The BBC has to make amends, particularly to assuage the hurt sentiment of a very large number of people.
"We understand the free press - they are welcome to explain and to challenge as long as it is fair and above the belt. Can this pass as acceptable journalism? The BBC has a global reputation. We expect the BBC to make amends."
The diplomats criticised a sequence in the programme involving banners being put on trains, saying "British IT is good for your company" and "Eat English muffins". The messages became obscene when the carriages parted and they were ripped apart.
Jeremy Clarkson also fitted his Jaguar with a toilet in the boot, saying: "This is perfect for India because everyone who comes here gets the trots."
Last year, a Top Gear episode that caused controversy by featuring jokes about Mexicans was cleared by broadcasting watchdog Ofcom, despite attracting 157 complaints and an objection from the Mexican ambassador.
Ofcom noted at the time that Top Gear was well known for its "irreverent style and sometimes outspoken humour".
Clarkson is one of the BBC's highest-paid stars, and this week it emerged that he made £2.14m last year from his involvement in a company that exploits Top Gear commercially around the world.