On the Breakfast programme this morning Henry asked the Prime Minister about who he was looking at to replace Sir Anand Satyanand when his five year term ends this year.
"Is he even a New Zealander?" Henry asked about Sir Anand.
Mr Key said he was.
"Are you going to choose a New Zealander who looks and sounds like a New Zealander this time... Are we going to go for someone who is more like a New Zealander this time?"
Race Relations Commissioner Joris de Bres said Henry showed appalling ignorance.
"The implication of his comments seems to be that anyone of Indian or Asian descent cannot be a New Zealander and should not hold public office unless they `look or speak like one'," he said.
"I am intrigued to know what Paul Henry's definition is of looking and speaking like a New Zealander, given that New Zealand's population includes 650,000 people of Asian descent and many people of different ethnicities who speak English with a wide variety of accents."
A spokeswoman for TVNZ said there had been two complaints this morning.
However; "The audience tell us over and over again that one of the things they love about Paul Henry is that he's prepared to say the things we quietly think but are scared to say out loud," she said.
"The question of John Key is the same, we want the answer but are to scared to ask."
Mr de Bres found TVNZ's response concerning.
Regular presenter on gadget and technology issues, Ben Gracewood, has quit his role with Breakfast over the comments.
"I've had enough... I love our multicultural, vibrant country to death and anyone who would like to foment racism simply for a laugh or to get more viewers can take a jump."
He said he would be happy to return if Henry left the show.
Henry told the Stuff website he stood by what he said and "some people are very easily offended".
Sir Anand is New Zealand's first Governor-General of Indian and Pacific ancestry. He had a lengthy career as a lawyer, judge and ombudsman before taking up the job in August 2006.
He was born and raised in Auckland, attending Richmond Road School in Ponsonby, and Sacred Heart College in Glen Innes. His parents were born in Fiji and migrated to New Zealand, his grandparents were born in India and had migrated to Fiji.
Sir Anand is in India for the Commonwealth Games. His public affairs manager, Antony Paltridge, said: "We won't be dignifying that sort of comment with a response."
Labour leader Phil Goff said Sir Anand was as much a New Zealander as he was.
"Anand is a person that grew up in this country, understands this country as well as anybody... a fantastic Governor-General."
Henry has previously called Scottish singing star Susan Boyle "retarded" and on another occasion mocked the facial hair of one of his female guests; anti-nuclear campaigner and Greenpeace worker Stephanie Mills.
The Broadcasting Standards Authority upheld complaints in those cases.
