Susanna Pressel, a Labour city councillor for West Oxford said, "I was horrified to see this strange church persist with its antediluvian policy of opposing women priests."
The Diocese of Oxford said the church, St Barnabas and St Paul with St Thomas the Martyr Parish was one of a small proportion of parishes that believe on theological grounds that their vicar should be a man.
Ms Pressel continued: "St Barnabas Church would surely be a much better place if it had a woman priest. Some of the local people who have left might return, but we now see that the dinosaurs have won yet again."
The Diocese of Oxford said that out of the 46 churches in the Oxford Archdeaconry, five seek male headship.
Employers in the UK must comply with equal opportunities employment laws, which make discrimination due to gender, sexual orientation or ethnicity unlawful, but the church is exempt from this when there are theological grounds.
A Diocesan spokesperson said, "The Church of England is fully committed to the ministry of women and men at every level of its ordained leadership."
"While many of those worshipping in the parish are very supportive of women in ministry, and St Barnabas has female licensed lay ministers and a Deacon who is a woman, a proportion of their congregation have said that they cannot remain within that church on theological grounds, unless the incumbent is a man."
"The parochial church council asked the Bishop of Oxford and patrons to respect those views in the interests of church unity and agree only to appoint a man to this vacant incumbency. They are permitted to do this under current legislation, and the Bishop and Patrons are bound by that same legislation to respect their view. Everyone is committed to the appointment of a Vicar for St Barnabas with St Thomas who will be an excellent parish priest for that parish, and be an asset to their local community."
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