The 31-year-old Williamson earned a second yellow card for a reckless barge on Leicester striker Jamie Vardy after 62 minutes, prompting an angry response from interim manager John Carver.
Carver said that he thought the defender had done it deliberately after Newcastle had suffered an eighth straight league defeat.
"I would like to take this opportunity to apologise to the Newcastle United supporters and my fellow team mates for getting sent off during yesterday's game,” Williamson said in a statement on the BBC website on Sunday.
"There was absolutely no intent to hurt Jamie Vardy or get sent off, but I recognise that I should have been more composed and measured in my determination to win the ball. Unfortunately, it did the one thing that I never wanted to do, and that was to hurt the team.
"Moving forward, I will do whatever I can to help my team mates prepare for the next two games and fight to get back into the team when I'm next available.
"I've had the honour to play for Newcastle United for five years now, and I can genuinely say that I feel privileged to play for this football club. I would never do anything intentionally to hurt the team or its supporters."
Newcastle are 15th in the table, two points clear of the relegation zone with three games remaining.
Carver, who has won two out of 16 Premier League games as manager after replacing Alan Pardew in January, has come under increased media fire after openly criticising Williamson.
Former Newcastle striker Michael Owen, who was relegated to the Championship with the north-east club in 2009, believes Carver’s comments could do more harm than good.
"Amazing quotes from Carver. Fans may like that type of honesty but as a manager, I'm not sure you can say things like that,” Owen wrote on Twitter. “I guess we haven't heard the end of this one."
Ex-England defender Martin Keown is also unsure Carver is up to the demands of management.
"When you're in a position like Newcastle are in, you have to stick together, but John Carver is digging players out,” Keown told the BBC.
“He needs guidance on how to behave because he is taking players apart.”
(Reporting by Michael Hann; editing by John O'Brien and Toby Davis)
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