The Conservative Anglican group was reacting to Rt Rev Chamberlain's admission in a newspaper that he was gay.
He said he was following Church teaching by remaining celibate and the Church of England said he had broken no rules.
Most Rev Justin Welby, the Archbishop of Canterbury, said Mr Chamberlain's sexuality is "completely irrelevant".
"I am and have been fully aware of Bishop Nick's long-term, committed relationship," he added.
However, in a statement, GAFCON said Archbishop Justin was wrong.
"There are aspects of this appointment which are a serious cause for concern for biblically orthodox Anglicans around the world, and therefore we believe that this appointment is a major error," it said.
"This news story will be seen by many orthodox Anglicans as yet more evidence that the clear biblical teaching in the Church of England on sin and salvation, human personhood, singleness, sex and marriage is being eroded and conformed to the values of secular society.
"While we pray for Bishop Chamberlain, our confidence in the processes by which he was appointed in are sadly, further diminished."
Bishop Nick was forced to go public about his sexuality because of threats by a Sunday newspaper to "out" him.
"It was not my decision to make a big thing about coming out," he said.
"People know I'm gay, but it's not the first thing I'd say to anyone. Sexuality is part of who I am, but it's my ministry that I want to focus on."
He said the Church was aware of his sexual orientation when he was appointed in November last year.
"I was myself. Those making the appointment knew about my sexual identity," he added.
GAFCON accepted Bishop Nick has "has many gifts as a leader and pastor" but warned he would "exacerbate" divisions in the Church over the treatment of gay people.
It said guidance that being gay was acceptable as long as a clergyman or woman is celibate were creating confusion.
"It is not modelling a helpful way to live, given the reality of our humanity, and temptation to sexual sin," it said.
A Church of England spokesman said: "The Church has said for some time that it would be unjust to exclude from consideration for the episcopate anyone seeking to live fully in conformity with the Church's teaching on sexual ethics or other areas of personal life and discipline.
"Whilst Bishop Nick's appointment is notable in the gifts and talents that he brings to the episcopate, it is wholly consistent and unexceptional in other regards given the testing of that call by those responsible for the selection process in each case."
Bishop of Lincoln Christopher Lowson said: "I am satisfied now, as I was at the time of his appointment, that Bishop Nicholas fully understands, and lives by, the House of Bishops' guidance on Issues in Human Sexuality.
"For me, and for those who assisted in his appointment, the fact that Bishop Nicholas is gay is not, and has never been, a determining factor.
"I understand that in some parts of the church, locally, nationally and internationally, this news will be challenging. My prayer for the church is that we will continue to seek to work together to understand difference with respect and dignity; to embrace and nurture our diverse gifts as disciples of Jesus Christ and in the service of God and neighbour; and to enrich and enable fulfillment in the lives of all God's people, whatever their background, race, faith, gender or sexuality."