The SDLP's Alasdair McDonnell said the social development minister failed to uphold the rule of the law and the ministerial code of conduct, on 24 September 2012.
Mr McDonnell was proposing the joint SDLP and Sinn Fein exclusion motion.
It said Nelson McCausland failed to condemn loyalist bands who breached Parades Commission rulings in north Belfast.
Mr McDonnell said the actions of the band showed "the ugly face of sectarianism".
He said that Mr McCausland had said the band was "simply playing a pop song" outside St Patrick's Church on Donegall Street in Belfast.
Mr McDonnell rejected the argument that the actions of the band had been a form of civil disobedience.
"This was not civil disobedience, this was a criminal act, criminal disobedience," he said.
Caitriona Ruane of Sinn Fein spoke to support the motion.
She said a determination of the Parades Commission had been deliberately breached by the band and called on the minister to apologise immediately.
Ms Ruane said it was the job of politicians to lead and "not to pander to sectarianism".
DUP leader and First Minister Peter Robinson said the motion was "disappointing" and "depressing".
"It is clearly a sectarian agenda with a sectarian tag team putting down the agenda," he added.
Mr Robinson said the SDLP and Sinn Fein "never believed that their motion would be accepted".
He accused the supporters of the motion of wanting to raise tensions ahead of a forthcoming parade.
"They choose a minister from north Belfast and they decide to demonise him," Mr Robinson said.
He said the Orange Order had taken all the steps to reduce tensions.
UUP leader Mike Nesbitt said he made no apologies for the DUP.
"They tend to tackle the man, or woman, rather than the issue," he said.
In a reference to the forthcoming Covenant Day parade on 29 September 2012, Mr Nesbitt said members should remember "the week that is in it".
He said his party would listen to the debate and then decide how to vote.
Stewart Dickson said the Alliance Party would support the motion.
He said the "standards of modern democracy must be upheld".
Mr Dickson said it was the role of ministers "not only to abide by the rulings of the Parades Commission but to uphold the rulings of the Parades Commission".
He called upon the minister to "apologise and withdraw his comments".
Traditional Unionist Voice MLA Jim Allister said the motion demonstrated that the house effectively had no control over some of the ministers of the house as the two largest parties had a veto.
Independent MLA David McNarry said the motion was an example of "pious hypocrisy".
Nelson McCausland began his reply to the motion by stating that he had not broken the ministerial code or the assembly code of conduct.
He said the motion was "much more to do with the desperate plight of the SDLP".
The minister said he had first heard about "the charade of this spurious motion" when sitting in an airport coming home from a meeting in Westminster concerning welfare reform.
He argued that the SDLP had been born out of civil disobedience, and said SDLP MLA Colum Eastwood had carried the coffin of a member of a dissident republican group.
"This was not just a funeral it was a paramilitary display," the minister added.
He gave examples of occasions when, he said, republican flute bands had played music outside Protestant churches.
Mr McCausland said: "The way forward is the way of mutual respect."
Gerry Kelly of Sinn Fein wound for the motion.
He said that perhaps the leadership of the DUP needed to take Mr McCausland to one side to tell him what he did was wrong "and to catch himself on".
The motion was defeated on a cross-community vote.