The Rt Revd Philip North turned down the post following objections to his stance against ordaining women priests.
He said he had withdrawn for "personal reasons" but added the "attacks" against him were "extremely hard".
The archbishops said the incident had raised concerns within the Church.
When he stepped down, Mr North said: "It is clear that the level of feeling is such that my arrival would be counter-productive in terms of the mission of the Church in South Yorkshire and that my leadership would not be acceptable to many."
In a joint letter, the Archbishop of Canterbury Justin Welby and the Archbishop of York Dr John Sentamu, said the events surrounding the appointment raised questions about the Church's 2014 commitment to the consecration of women bishops.
They said: "The recent events surrounding the nomination of Bishop Philip North as Bishop of Sheffield, including his withdrawal from the process, have understandably raised great concern amongst many in the Church of England.
"The status of the House of Bishops' Declaration, of June 2014 has been questioned by some and its meaning has also been challenged.
"We have therefore written to Sir Philip Mawer, the Independent Reviewer under the Declaration on the Ministry of Bishops and Priests, (Resolution of Disputes Procedure Regulations) 2014, to address the concerns that have arisen in the Church following these recent events."
It asks Sir Philip to examine five areas, including what efforts have been made to educate clergy and the public about the declaration and whether the nomination of Mr North was consistent with its terms.
The archbishops also used the letter to call on all those in the Church to "pray for those with whom they disagree, demonstrate mutual love and overcome those those things which in ourselves we find so divisive".
The Archbishops of Canterbury and York have called for an independent inquiry into the circumstances surrounding the failed appointment of a new Bishop of Sheffield.