“I'm sorry our BBC Panorama on Tommy Robinson wasn't broadcast,” Mr Sweeney wrote on Twitter, referring to a planned edition of the programme that was not broadcast about the English Defence League.
It was cancelled after Robinson, whose real name is Stephen Yaxley-Lennon, covertly filmed Mr Sweeney and made a rival “documentary”.
Mr Sweeney added: “So after 17 years I can finally say these are not the views of the BBC but he's a complete c**t.”
Robinson was the focus of a documentary due to be broadcast earlier this year, after he was imprisoned for contempt of court last year.
During production, one of his former aides secretly filmed Mr Sweeney in a Cambridge pub after they offered to give interviews to Panorama.
The footage showed the journalist calling the BBC a “stupid and irritating organisation”. He also claimed that managers had previously attempted to make him redundant.
After ordering a significant quantity of alcohol, Mr Sweeney said he was “on expenses”.
He also suggested he was from a “scumbag Irish background”, before calling a counter-extremist researcher a “bloody woofter”.
Robinson played the footage to Mr Sweeney and told him “you’re finished” after arranging to conduct an interview with Panorama.
“If you think I’m finished you’ve got another thing coming and I can give you that for free,” Mr Sweeney replied.
Robinson's video was compiled into a film called Panodrama and played at a protest by his supporters in Salford Quays in February.
The far-right activist later claimed credit for Mr Sweeney's departure from the BBC in a message to supporters on the Telegram messaging app.
The BBC previously rejected Robinson’s claims that its journalism was “faked or biased” and said all programmes adhere to strict editorial guidelines.
“Some of the footage which has been released was recorded without our knowledge during this investigation and John Sweeney made some offensive and inappropriate remarks, for which he apologises,” a spokesperson said earlier this year.
The National Union of Journalists condemned Robinson and “far-right thugs” for attempting to intimidate BBC reporters including Mr Sweeney.
Mr Sweeney said he “paid for all the drinks” after announcing his departure from the corporation.
He said it was “high time to make trouble elsewhere” and said he would travel to Malta in relation to a book on murdered investigative reporter Daphne Caruana Galizia.
“I remain an old school reporter, up for the right kind of trouble,” he added. “I'll be back.”
Mr Sweeney worked at The Observer before joining the BBC in 2001, working on investigations including the wrongful imprisonment of mothers over cot deaths, Scientology, North Korea and Russian influence on the US election.
In an email to staff, the BBC's head of current affairs Joanna Carr praised his "courageous and award-winning journalism", adding: "John's investigations have made a significant contribution to BBC journalism and we wish him all the best for the future."