It follows the resignation of the Archbishop of Canterbury, Justin Welby, last week after a review found "he could and should have" reported a prolific paedophile to the police.
Dame Jasvinder Sanghera claimed Mr Cottrell had ignored "separate complaints" about other alleged abusers, which he strongly denied.
She said: “Anybody in a senior leadership position has to have trust and transparency, and you certainly need to have the trust and confidence of victims and survivors harmed by members of the clergy.”
“It’s not just my call. This is call of many individuals who have been harmed by members of the clergy," she told BBC Radio York.
Dame Jasvinder was a survivor advocate on the Church of England’s Independent Safeguarding Board until the panel was sacked in June 2023 when the church said relations between board members and bishops had "broken down".
The Church of England said Mr Cottrell had always taken safeguarding very seriously, particularly his commitment to victims and survivors.
It said: “He commends Dame Jasvinder's personal work in this important area. But, it is not true to say that these victims and survivors have been ‘ignored’.
“He believes the future of church safeguarding needs independent structures and scrutiny while ensuring that it also remains everyone's responsibility, as highlighted in the recent Makin review.”
Mr Welby’s resignation followed a report which found he "could and should" have reported abuse of boys and young men committed by John Smyth to police in 2013.
Smyth was accused of attacking dozens of boys, including those he met at Christian camps, in the UK in the 1970s and 1980s.
The Archbishop of York is the second most senior bishop in the country after the Archbishop of Canterbury.
Mr Cottrell has held the position since July 2020 and oversees dioceses across the north of England.