Lord Carey, who was the Archbishop of Canterbury from 1991-2002, told the investigation that he did not remember Tudor’s name.
In his resignation letter, sent on 4 December, he wrote: “I wish to surrender my Permission to Officiate.”
“It has been an honour to serve in the dioceses of London, Southwell, Durham, Bristol, Bath and Wells, Canterbury and finally Oxford,” Lord Carey said
“I am in my ninetieth year now and have been in active ministry since 1962 when I was made Deacon and then Priested in 1963.”
The current Archbishop of York Stephen Cottrell is resisting calls to resign in relation to the same investigation.
The BBC found that from 2010, when he was made Bishop of Chelmsford, Mr Cottrell was made aware of the allegations of abuse against Tudor.
Mr Cottrell also knew that Tudor was working as a priest under a provision that he could not be left alone with children and that he had made a large payment to an alleged victim.
But under the then Bishop Cottrell, Tudor remained in post and was made Honorary Canon of Chelmsford Cathedral.
Tudor was only suspended in 2019 when police opened a new investigation into him.
Mr Cottrell said he was “deeply sorry” that action had not been taken earlier but that he had “inherited” the situation.
He added there were no legal grounds to take alternative action.
But some of those affected by the case, including victims of Tudor, have been critical of Archbishop Cottrell’s response to the investigation.
One, Rachel Ford, told the BBC Tudor was “very touchy-feely” with her at school and that Mr Cottrell’s assertion showed “a complete lack of empathy for his victims”.
Mr Cottrell is weeks away from taking charge of the Church of England following the resignation of another Archbishop of Canterbury Justin Welby.
Mr Welby stepped down last month over his management of a separate sex abuse case.
Additional reporting by Harry Farley
