"My comments were by no means the worst comments made," he told the radio host.
"It was a fun sort of thing, everyone laughed their heads off."
Williams was given an award for the year's best celebration at the annual caddie awards a tongue-in-cheek reference to his claim that his current employer Adam Scott's victory at the Bridgestone Invitational in August was "the best win of my career".
Williams was working with Scott after being sacked as Woods' bagman despite them having won 13 majors together.
Asked what that celebration was all about, the 47-year-old New Zealander said: "It was my aim to shove it right up that black a***hole."
Williams refused to tell the Herald on Sunday whether he had any concerns the remarks would affect his career and initially defended the comments.
"It was a fun night. It was a funny award. There were a lot of other funny awards and a lot of funny things said. If you can't have a bit of fun, what is the world coming to?" he said.
When asked if he used the words "black a***hole" he initially denied using the racially motivated language: "I did not say those two words."
However, he refused to clarify exactly what he did say on stage at the event, attended by more than 100 top players, caddies and golfing officials.
"Things get taken out of context," he said.
It is likely Williams thought the event was private and comments would not be reported by media, but several websites reported them within minutes.
As the media storm grew yesterday, Williams issued a statement on his website.
"I apologise for comments I made last night at the Annual Caddy Awards dinner in Shanghai. Players and caddies look forward to this evening all year and the spirit is always joking and fun. I now realise how my comments could be construed as racist. However, I assure you that was not my intent. I sincerely apologise to Tiger and anyone else I have offended."
Woods has not reacted to the comments but his agent, Mark Steinberg, revealed he was with Woods in Australia when he heard the news.
"It's a regrettable comment and there's nothing Tiger can do or say. He's just going to move on."
New Zealand race relations commissioner Joris de Bres said it was an "appalling" remark.
One caddie said: "Never have you been in a room and seen so many jaws drop at the same time. We knew he was an idiot but we didn't know he was a racist idiot. I was standing next to a European Tour official who said, 'Thank God he is not on our tour'."
It is believed Williams had no intention of attending the function until he was persuaded to do so by Scott.
Williams was on Scott's bag throughout the third round of the tournament yesterday and the Aussie backed his man.
He said the comments at the private party were not meant to leave the room and that Williams' apology should be accepted.
However, former New Zealand No1 Greg Turner said he did not think Williams could survive the media backlash, particularly in the United States.
"I can't see this ending well. There's some things you can't get away with and this is one of them. I have been trying to think of a context that might have been acceptable but can't come up with anything. I think this is a pretty big one," he said.
- Additional reporting: Agencies
