Newly-selection Bishop of Sheffield Right Reverend Philip North was urged last week to turn down the appointment over concern of "public damage to the Church".
Responding, Dr John Sentamu told the Yorkshire Post: "Women clergy in the Diocese of Sheffield will not only be accepted, but will be encouraged, inspired, and furthered in their ministry by their new Diocesan Bishop.
"However, there remain those who question the integrity both of the agreement reached by the Church of England, and of Philip North himself. And that simply won't do."
Archbishop Dr John's comments came after Bishop Philip's appointment wa criticised by the Dean of Christ Church in Oxford who called upon him to reject his nomination or withdraw his membership of a group which rejects female clergy.
Very Rev Martyn Percy said: "Sheffield is a go-ahead, vibrant, progressive city, with cutting-edge universities and research-led industries," Percy wrote.
"The public will neither comprehend nor welcome this rather fogeyish sacralised sexism of the religious organisation - known simply as 'The Society'."
Bishop Philip has previously staid he is "determined to be a bishop for all and develop the ministry of all clergy - male and female."
Archbishop Dr John went on to say: "This debate matters not just because of Philip's own integrity - which remains unimpeached in the midst of debate - but also because of the nature of the Church of England itself, which reached a resolution in 2014 that enabled people who disagree on fundamental issues to continue to remain together as members of one Church bound together in Jesus Christ.
"Philip brings tremendous energy for mission and ministry. He is a disciple of Christ, full of the Holy Spirit, grace and insight. We look forward to welcoming him to our North East region group of bishops."