Fury, whose homophobic and sexist comments had led to more than 130,000 people signing a petition calling for the BBC to remove him from its shortlist, told the crowd, featuring many of Britain's greatest sportsmen and women: "I've said a lot of stuff in the past, none of it with intentions to hurt anybody.
"It's all a bit tongue in cheek, it's all fun and games for me. I'm not really a serious type of person. Everything's happy go lucky with Tyson Fury.
"If I've said anything in the past that's hurt anybody, I apologise to anybody's who's been hurt out there. It's not my intention to do that."
Up to 30 gay and equal-rights campaigners held banners and chanted 'anti-woman, anti-gay, Tyson Fury go away' outside the arena before the Belfast ceremony.
Murray won the award for the second time in three years after his inspirational performances guided Britain to their first Davis Cup triumph in 79 years.
The 28-year-old, who joins boxer Henry Cooper and racing drivers Damon Hill and Nigel Mansell as the only double winners of the event, held up his 'Personality' award and joked: "I didn't expect this.
"A friend sent me a message the other day with an article saying 'Andy Murray is duller than a weekend in Worthing' which I thought was a bit harsh -- on Worthing," the Scot said of the southern England seaside resort.
Runner-up was rugby league player Kevin Sinfield and in third place was world heptathlon champion Jessica Ennis-Hill.
(Reporting by Ian Chadband; Editing by Tony Jimenez)
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