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The 44-year-old Italian was asked on several occasions at a news conference on Tuesday if he supported fascism and was clearly angered by the questioning, warning that he may not speak to media who continue to quiz him on the subject.
During the press conference, Di Canio refused to expand on a 2005 interview with an Italian news agency when he said he was "a fascist, but not a racist".
His appointment has led to the resignation of the club's vice-chairman David Miliband, Labour MP for South Shields and former foreign secretary.
Anti-racism group Kick It Out expressed concern about the far-right in football and said Di Canio should make his position clear.
A statement issued on Tuesday evening said: "It is not part of Kick It Out's remit to sanction the selection of staff of football clubs. However, football clubs have a responsibility to ensure that their employees demonstrate a commitment to anti-discrimination and equality of opportunity. It may be in the interest of both the club and Mr Di Canio to acknowledge a full and frank commitment to these policies.
"It is all too easy for positive progress and engagement to be compromised by inappropriate statements, and both Sunderland supporters and members of its local community will be looking for reassurance on these points.
"When fascism is referenced within the context of English football, there is a wider concern. The spectre of the rise of far-right groups in some parts of Europe cannot be allowed to undermine the progress made in the game domestically, and the increasing number of incidents involving both neo-Nazi and Ultra groups which have happened in places like Italy and Greece, must not be replicated here.
"Kick It Out and its partners have held concerns for some time about similar groups operating in the UK. Now is the time to reinforce this vigilance."
Di Canio however blasted the furore over his appointment as Sunderland manager as "ridiculous and pathetic" and had a warning for those in the media who continue to question him on the subject.
He said on Tuesday morning: "I don't have to answer any more this question. There was a very good statement from the club, (with) very, very clear words that came out from me.
"My life speaks for me so there is no need to speak any more about this situation because it's ridiculous and pathetic.
"I can't every two weeks, every two months, every 10 months answer the same questions that are not really in my area.
"We are in a football club and not in the House of Parliament.
"I'm not a political person, I will talk about only football. People that respect others have to be straight and I don't want to give a chance any more to someone for example who comes to a press conference and talks about something that is not football."
Di Canio has been pictured making a fascist salute to Lazio fans in the past and the Durham Miners' Association has asked the club to return a symbolic banner which is kept at the Stadium of Light if Di Canio remains in his post, describing his appointment as a "betrayal and a disgrace".
Asked on Tuesday if he would meet the association, Di Canio said: "I don't understand this problem. I am a manager, I'm a normal man, I'm a family man.
"The words sometimes fly away, what counts in life is the facts, who you are, what you are doing, what you did in the past so I think that should be enough and that they will be happy one day."
He also insisted the controversy would not impact on the squad, who are only a point off the relegation zone.
He said: "It's not a distraction, they are footballers and only because the club has changed the manager is it a big shock."
Di Canio himself said, like at Swindon previously, he would win over those fans unhappy at his appointment.
He said: "It was the same for a different reason at Swindon. There were many, many people who could not believe that Di Canio was the right manager for the club.
"After two months, there were 9,000 people at the end of the game clapping their hands for my players and singing my name.
"It is more important the Sunderland fans sing my players' names as they need the extra lift. They will then be happy, probably, as they will think they were wrong and now we have to keep Di Canio forever.
"The press call me the mad Italian but I would confidently bet everything I have on Sunderland remaining in the top flight.
"When I got the call, I felt fire in my belly. I would have swam to Sunderland to take the job."
Sunderland's new partnership with The Nelson Mandela Foundation is not under threat following the appointment of Di Canio, the organisation confirmed on Tuesday.
A statement said the Centre had met with the club and Di Canio "to discuss the public debates around Sunderland's new coach" and that the club has reaffirmed its commitment to anti-racism and equality.
Meanwhile, the BBC said it did not pay a fee to Di Canio to write six columns for the BBC Sport website between August 2012 and last month.
