The heavyweight world champion was seen making alleged homophobic, sexist and anti-Semetic comments during the 57 minute clip.
He was also shown giving views on bestiality, paedophilia and women.
His comments led to calls for him to be banned from boxing.
It's not the first time he's got himself into trouble over comments made outside of boxing.
Last year before beating Wladimir Klitschko to become world champion, 100,000 people signed a petition for him to be removed from the BBC's 2015 Sports Personality of the Year shortlist because of previous comments on homosexuality, paedophilia and women.
He later claimed his comments had been taken out of context.
Speaking about the most recent comments, Mr Fury said: "I apologise to anyone who may have taken offence at any of my comments. I said some things, which may have hurt some people, which as a Christian man is not something I would ever want to do.
"Though it is not an excuse, sometimes the heightened media scrutiny has caused me to act out in public. I mean no harm or disrespect to anyone and I know more is expected of me as an ambassador of British boxing and I promise in future to hold myself up to the highest possible standard.
"Anyone who knows me personally knows that I am in no way a racist or bigot and I hope the public accept this apology."
Tyson Fury is currently preparing to defend his world championship belts in a rematch with Wladimir Klitschko, which takes place in Manchester in July.