Rt Rev Philip North said the decision to refuse the nomination was made "with regret and sadness" but he believed his arrival "would not be acceptable to many" and be "counter-productive" for local mission.
In a statement released on Thursday night, he said: "The highly individualised nature of the attacks upon me have been extremely hard to bear.
"If, as Christians, we cannot relate to each other within the bounds of love, how can we possibly presume to transform a nation in the name of Christ?"
Bishop Philip's traditional views on women leadership in the Church has put him at odds with others within the Church of England which allows women to become vicars and bishops.
In open letter on Wednesday, Sheffield Heeley MP Louise Haigh (pictured above) asked that Bishop Philip meet with her over what she described as his "troubling" views.
The Labour politician wrote: "Your traditionalist views pose many questions about how your leadership in Sheffield will work practically in relation to existing women clergy, men who have been ordained by women, women who are in the process of becoming clergy and congregations who come from a completely different tradition within the church."
Bishop of Doncaster Rt Rev Peter Burrows said he was "deeply and personally saddened" to learn Bishop Philip had refused his nomination to become the next Bishop of Sheffield.
He said: "This has clearly been a difficult and painful journey for the Diocese over the past few weeks as it has also been for Bishop Philip.
"There will be much to reflect on and there will be time to consider what lessons may be learned over the coming weeks and months."
Downing Street said that Archbishop of York Dr John Sentamu will "in due course submit the name of an alternative candidate for this diocese".