"It was the most satisfying."
Williams said he and Woods had not spoken since he was fired in July.
Williams said he was overcome with emotion after Scott birdied the 18th hole at the $10 million Bridgestone Invitational in Ohio.
"The people were constantly giving you a cheer and a clap every time you go to a tee or a green, just like they do with players.
"Then going down to the last hole was something that I'll never ever forget. They were just chanting out: 'Stevie, Stevie, Stevie'."
Golfers and sports commentators criticised Williams for his comments after the tournament - saying caddies were not meant to talk to media and he stole Scott's limelight.
European golfer Chris Wood called his actions shameful because he did not mention Scott when he talked about the pair's win.
"The last thing I expected was to have a microphone shoved in my face. I mean, they don't do that and it's never been done before and it took me by complete surprise," said Williams.
"I did make the mistake of not mentioning Adam's name and that wasn't intentional but in all the emotion pouring out of you, that's what happened which can happen in the heat of the moment sometimes.
"I understand that - that caddies don't speak to the media. I've caddied for 33 years ... But there was obviously a lot of interest in what had just occurred, so at that time I thought it was appropriate so answer some questions."
Williams, 47, said he had apologised to Scott. "He totally understood. He could see just how fired up I was on the final hole and he could see what the moment meant to me," he said.
"Did I make a mistake? Yes, but we all make mistakes ... Obviously I've upset the players and my fellow caddies, so I apologise for that but that's what happens when your emotions run high."
