Fury was crowned world champion following his win against Wladimir Klitschko in Dusseldorf at the end of last month. After this he said: "I'm not sexist. I believe a woman's best place is in the kitchen and on her back. That's my personal belief. Making me a good cup of tea, that's what I believe."
In a separate interview with the Mail on Sunday, he said: "There are only three things that need to be accomplished before the devil comes home: one of them is homosexuality being legal in countries, one of them is abortion and the other one's paedophilia.
"Who would have thought in the '50s and '60s that those first two would be legalised?"
A petition to remove Fury from the BBC SPOTY competition had received nearly 100,000 signatures at the time of this story's publication.
Fury has responded to the backlash, tweeting:
I'm a mans man, who ever don't like can go hoot there selfs, I follow Jesus, & I believe everything he says!!! If this offends u grow up!— Gypsy King (@Tyson_Fury) December 6, 2015
LGBT charity Stonewall said it was "shocked" and "disappointed" by the BBC's decision to keep Fury in the list.
Ruth Hunt, the charity's chief executive, said: "Sportsmen and women are role models, and their views and opinions on everything, from fashion to politics, are respected and repeated by many of their fans. Homophobia has long been a barrier to the participation of lesbian, gay, bi and trans people in sport.
"It's often assumed that people of faith can't be LGBT or respect LGBT people. By suggesting his homophobia is rooted in his faith he has helped perpetuate this damaging myth."
She concluded: "Sadly, some role models are actively working against us. Sadder still, they're being celebrated."
A statement from the BBC said: "Tyson Fury’s victory and subsequent crowning as world heavyweight champion was widely covered by media outlets and his inclusion on the shortlist is similarly recognition of his sporting achievement – it is not about endorsing his personal views. It will be for the public to decide who will be crowned Sports Personality of the Year 2015."
Fury is one of 12 contenders for the SPOTY award, which will be awarded on Sunday 20 December.
Yesterday it was announced that Fury has been stripped of his IBF title just two weeks after winning it. The boxer was supposed to fight the IBF challenger Vyacheslav Glazkov but instead signed a rematch clause with Wladimir Klitschko. He retains the WBA and WBO versions of the heavyweight title.
This article was updated with further information.