"What might have been something that was aimed at poorly-designed humour has ended up embarrassing New Zealand and for that I'm quite regretful," Key told reporters.
The presenter, Paul Henry, mocked the name of Delhi Chief Minister Sheila Dikshit and added "it's so appropriate because she's Indian", prompting an official complaint from India over the "racist" and "unacceptable" comments.
Key ruled out making a personal apology, saying Wellington's envoy in New Delhi had already condemned the remarks on his behalf after being summoned to a meeting Thursday with India's External Affairs Minister S.M. Krishna.
"The reason I won't is because in a sense that's already happened, it doesn't matter if the words don't come out of my mouth," he said.
In his apology, New Zealand High Commissioner Rupert Holborow said Henry's comments were "culturally insensitive, inappropriate and vulgar" and did not represent the view of the New Zealand government or its people.
New Zealand's Foreign Minister Murray McCully also sought to defuse India's anger, describing Henry's remarks "gratuitous and insulting" but admitting he was powerless to act against the presenter.
He said Henry's remarks on a live breakfast show last Friday were a regrettable abuse of freedom of speech.
While the presenter made the remarks on state-owned television station TVNZ, McCully said he wanted to make it clear to India that under New Zealand law the broadcaster operated independently of government.
McCully said he would contact the Indian government to assure it "the comments were the actions of one person, made in a country in which freedom of speech is an important foundation principle".
"Any action against Mr. Henry is entirely a matter for the company, or for the Broadcasting Standards Authority," he added.
Opposition leader Phil Goff said Key's response to the issue had been weak, arguing the prime minister should have immediately called Indian officials to apologise.
"The prime minister had the opportunity to be strong and unequivocal on something that surely he believes is wrong and offensive and he could kill the issue. He should have done it," Goff told Radio NZ.
He said the row threatened to hurt New Zealand's relationship with New Delhi in the same way that Australia's ties with the country were strained last year after a series of assaults on Indian students in Melbourne and Sydney.
Henry's remarks about Dikshit initially passed largely unnoticed until TVNZ included the footage in its website's highlights section. Since then, it has been viewed more than 300,000 times on youtube.com.
Henry is already serving a two-week suspension over comments on Monday questioning whether Governor General Anand Satyanand, who was born in Auckland to Indo-Fijian parents, was a proper New Zealander.
The controversial host made an on-air apology shortly before his suspension was announced but invited further controversy by referring to himself as "half-gyppo" (gypsy).
The owner of one of New Zealand's largest supermarket chains, Progressive Enterprises, said Henry's offensive behaviour could affect its advertising on TVNZ, warning it would be monitoring closely when the host resumes broadcasting on October 18.
The New Zealand Commonwealth Games team's chef de mission, Dave Currie, said Henry's remarks were extraordinarily disappointing and an unwelcome distraction for the country's athletes as they tried to concentrate on competing in the Delhi Games.
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