The Church of England is facing a rebellion from gay clergy over the question of same-sex partnerships.
Mr Macourt (left) and Rev Christopher Wardale in Newcastle
Some gay priests who want a civil partnership themselves have said they will refuse to give private details about their relationship.
The Church has reminded clergy that it would view those in civil partnerships as being in a sexual relationship.
Traditionalist Anglicans are calling on the Archbishop of Canterbury to carry out promised disciplinary action.
In recent correspondence, the Church of England reiterated its teachings that sexual relations should only take place within a heterosexual marriage.
The church is not enforcing its directive on lay members who have civil partnerships but is telling clergy they must promise that their relationships will be celibate.
BBC News religious affairs correspondent Robert Piggott said there are already signs of a rebellion from gay clergy.
He said the issue of homosexuality has already split the worldwide communion and now poses a new threat to unity at home.
Wednesday saw the first same-sex ceremonies take place in England and Wales.
One vicar, the Rev Christopher Wardale, 59, from Darlington, and retired Northumbria University lecturer Malcolm Macourt, 58, were among the first people to exchange vows in Newcastle this week.
But the couple have refused to talk about the nature of their relationship.
The Rev Paul Collier, meanwhile, wrote to his Bishop this weekend saying he would be registering a same sex partnership and would not be making an promises as to its nature.
"I think the church has got itself into a mess over the issue," he told BBC News.
"There is no reason why anyone should worry about any aspect of any person's private life or any priest's private life unless there is some kind of scandal."
Traditionalists such as Canon Chris Sugden, of the Anglican Mainstream coalition, have voiced their opposition.
"The Archbishop of Canterbury has promised that the Church and the House of Bishops would exercise discipline," he said.
"If they take no action in this case, then frankly, they will not be believed anymore throughout the Anglican Communion as to their word."