Henry asked Prime Minister John Key on the Breakfast programme this morning about who he was looking at to replace Sir Anand when his five-year term ends this year.
"Is he even a New Zealander?" Henry asked about Sir Anand - who was born and raised in New Zealand by his Fijian Indian parents.
"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?"
The comments prompted complaints to TVNZ, led to the resignation of a regular Breakfast presenter and were condemned by Race Relations Commissioner Joris de Bres as showing appalling ignorance.
Henry is no stranger to controversy but, unusually, moved to take the heat out of the situation by apologising this afternoon.
"I am aware that Sir Anand has made an outstanding contribution to New Zealand," he said in a statement.
"Anyone who knows anything about me will know I am a royalist, a constant defender of the monarchy and the role the Governor-General plays in our society.
"If my comments have personally offended Sir Anand, I regret it deeply."
Mr Key condemned the comments as "just plain wrong" but said he still intended to appear on the show each Monday morning.
"I don't think I personally can be held responsible for what comes out of Paul Henry's mouth," he said.
Asked whether he should have objected at the time - instead of joking about whether Henry wanted the job - Mr Key said he was taken aback by the comments.
"I wasn't going to get into critiquing what he meant and I was a bit surprised when he said it, so I just carried on."
TVNZ had received two complaints soon after the comments and Mr Key said it was up to the Broadcasting Standards Authority (BSA) whether any action was taken.
TVNZ earlier defended Henry, saying: "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.
"The question of John Key is the same, we want the answer but are too scared to ask."
Mr de Bres said the implication of Henry's comments was that anyone of Indian or Asian descent could not be a New Zealander and should not hold public office.
"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."
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 BSA upheld complaints in those cases.
