Veitch also admitted that he paid Kristin Dunne-Powell a sum of money afterwards for loss of income and the distress he had caused her.
However, reading a statement at a news conference in Auckland, he gave no specific details about exactly what happened on the night in question.
The Dominion Post reported this week that Ms Dunne-Powell was hospitalised and had to spend time in a wheelchair after the attack.
It said she did not report the incident because Veitch, who hosts the Radio Sport morning show and is the TV One News sports presenter, paid her $100,000.
Supported by wife Zoe Halford, Veitch said he deeply regretted what happened and had done so every day since.
He said he had agreed to let Ms Dunne-Powell come over to his house, but the pair had "a major disagreement'' after dinner.
"We argued for along time,'' he said.
"In the end, my frustration took over and I broke and lashed out in anger, something I will regret to the end of my days.''
Veitch took issue with some of the reports in the media this week, "but again no excuses''.
"It's the thing I will most regret in my life,'' he said.
"I have lived with the night on my conscience ever since and I will always do.''
While not offering any excuses, Veitch said his relationship with Ms Dunne-Powell had just ended when the incident occurred.
He was working seven days a week in two "stressful'' jobs, was "emotionally and physically exhausted'' and was taking medication for exhaustion.
"I was in the lowest ebb of my life and I needed help, but again it was inexcusable''.
Veitch said the reason he had not spoken of the incident until now was that he and Ms Dunne-Powell had made a confidentiality agreement.
"That agreement included payment to Kristin for loss of income and distress I caused her,'' he said.
"I had no wish to breach that confidentiality but because of the growing controversy and the positions I hold, and the standards of behaviour those positions demand, I felt it important to offer some words of explanation.''
Veitch said he undertook counselling after the event and that had enabled him to form his relation with his wife.
He had told her about the incident early on and she had been "completely supportive''.
He apologised to Ms Dunne-Powell, and employers the Radio Network and TVNZ for embarrassment caused, and also to the New Zealand public.
After his three-minute statement, Veitch declined to take questions.
Radio Network general manager talk programmes Bill Francis accompanied Veitch to the news conference but did not appear with Veitch as the latter spoke.
Francis had said earlier that there would be no comment from him or the company after Veitch's statement.
Yesterday, Francis said Veitch had taken leave for the rest of the week from his Radio Sport job.
TVNZ head of news and current affairs Anthony Flannery said Veitch had agreed to step down while the state broadcaster conducted a review into the allegations.
Chief executive Rick Ellis said the review began on Monday and there had been a number of conversations with Veitch and his legal representatives.
Ellis said there were significant and complex matters around the issue that restricted what could be said publicly.
"We are giving the situation our most serious consideration, and Tony's personal statement today will be taken into account by the review process,'' he said.
"Violence is a major issue in New Zealand and the profile of our on-air presenters is important to the public and to our organisation.''
Police this morning said they could investigate the assault allegations but it would be difficult without a formal complaint or evidence.
Following Veitch's public appearance this afternoon, Auckland police spokeswoman Noreen Hegarty said his statement would be reviewed, but police were unlikely to spring into action immediately.
"It's been two or three years so it's not something that's necessarily going to warrant an instant response tonight in terms of policing.
"What we will do is assess his statement and make a determination about where to go from there.''
