David Lowe, 68, landed in hot water after airing an 82-year-old version of The Sun Has Got His Hat On featuring the racial slur.
The BBC Radio Devon broadcaster apologised to officials at the station, based in Plymouth's Seymour Road, but he said his words fell on deaf ears.
Boss Emma Clements reportedly told Mr Lowe while he had "properly" dealt with his mistake, she thought it would be best for him to step aside.
After his exit, Mr Lowe claims he was asked to say he was pursuing other interests. But he refused and made the reasons public through his website.
The BBC offered him his job back but he said he declined because his departure caused him so much stress and caused a flare-up of dystonic tremor, a medical condition he suffers from.
"It's going to take quite some time for me to get back to where I was before this N-word issue blew-up ... if I ever do," he wrote on his site.
A BBC spokeswoman said the corporation admitted the discussion about Mr Lowe's future "could have been handled better" and said the "door remained open" should he want to return.
She added: "His offer to resign was agreed locally but once escalated it was decided his resignation should not have been accepted … we would like to thank him for his many years presenting on Radio Devon and wish him well for the future."
Hundreds of people have taken to social media to offer Mr Lowe their support.