The fiery Italian took over from O'Neill at the Stadium of Light in March, but he was sacked in September after the club picked up only one point from their first five games this season.
New Republic of Ireland boss O'Neill branded Di Canio 'a charlatan' earlier this week, but the former Italy international insisted he saved a club that was "already sunk".
Speaking in an exclusive interview with Sky Sports News, Di Canio said: "I don't know if he knows the meaning of this word charlatan. Probably I can teach him, even if I am not English.
"I respect the opinion of manager Martin O'Neill but the fact that he spoke after six months, not straight away, that proves what kind of level he is. He is not very big.
"A charlatan is a manager who spends £40m to be a top 10 club and then sees the club sink into the relegation zone."
Di Canio also stood by his criticism of the condition of the Sunderland players when he took over, describing their fitness levels as "pathetic".
"I had players who told me they had cramps from driving the car," he said. "I had three players with injuries in the calf after 20 minutes of a game. Six different players with problems means they were not fit."
The Italian also hit out at Phil Bardsley after the Sunderland defender claimed Di Canio tried to destroy his career.
Di Canio accused Bardsley of "treason" after he posted a tweet mocking Sunderland following their home defeat to Fulham on the opening day of the season.
"At the beginning of the season, he made tweets celebrating the defeat of the club that pay him," Di Canio added. "A person at the club came to me and said 'we want to fine him', and I agreed.
"He was celebrating, that is the worst treason for the people next to you. It is clear that he tried to destroy his career on his own."
Di Canio also denied claims from striker Steven Fletcher that he did not want the players to smile or laugh.
He added: "Fletcher was never happy when I said, publicly, that after four years in England his record was two relegations, and twice 17th in the table.
"Maybe the players wanted a big screen with a comedy movie. When you have a training session you are there to improve. Maybe there is a moment when you have fun, and they probably had too much fun."
Di Canio has no regrets over his short tenure with Sunderland, and he remains keen to take another job in England with the "right club and the right people".
"The sacking made me stronger," he said. "I am my own worst critic. Sometimes a situation like this gives you a graduation. I think my level was too high for this situation.
"What doesn't kill me makes me stronger. I cannot wait to have another chance in the right place with the right people who let me work my way. Now I am a better manager than before, much better."
Di Canio also insisted that he was keen to take another job in the Premier League or Championship despite hinting he has fielded offers from clubs in Europe.
"Even if I have four requests around Europe, I don't go," he added. "I will wait until the time in England when there is no more space for me. Otherwise, I wait."
Sunderland made 14 new signings during the summer, but Di Canio insisted director of football Roberto De Fanti and chief scout Valentino Angeloni were responsible for all transfer business and claimed he would have wanted more British players.
"I think 80 per cent of the squad should be British footballers," he said. "I don't know why more didn't come, you would have to ask Roberto De Fanti and Valentino.
"They were given power by the chairman. I gave them my opinion, I gave them names, but not one came and I don't know why. The players that did come, I accepted, but obviously they weren't my first choices."