Breakfast show host Paul Henry released a statement that said: "I sincerely apologise to the Governor General, Sir Anand Satyanand, for any offence I may have caused."
During the TVNZ programme Monday morning, Henry asked Prime Minister John Key whether Anand was a New Zealander or not.
When Key told him that Anand was a New Zealander, Henry asked if he was going to pick someone who looked more like a New Zealander next time, New Zealand Herald reported Monday.
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 being appointed to the job in August 2006.
He was born and raised in Auckland. His parents were born in Fiji and migrated to New Zealand while his grandparents were born in India and had migrated to Fiji.
While apologising, Henry said: "I am aware that Sir Anand has made an outstanding contribution to New Zealand. 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."
Race Relations Commissioner Joris de Bres said Henry's comments were "extremely disrespectful" and "a sad commentary on his state of mind".
"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'," de Bres was quoted for saying.
The official added: "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 was an honoured guest at the colourful opening ceremony of the 19th Commonwealth Games in New Delhi Sunday evening.