At a press conference on Saturday, Senator Anning was hit with an egg by a 17-year-old. The Queensland senator then struck out twice with an open hand at the teenager, who was tackled to the ground by a group of the senator's supporters, including convicted criminal Neil Erickson.
His original comments attracted condemnation from fellow senators while members of the general public demanded he be excluded from Parliament altogether.
Senator Anning, who was elected to the Senate to fill the position of One Nation's Malcolm Roberts after the High Court found he was ineligible to be a member of the Upper House, is due to face voters at this year's election
In a joint statement, Senate leader Mathias Cormann and his Labor counterpart, Penny Wong, said a motion censuring Senator Anning would be debated when the Upper House resumes on April 2.
The motion will note that religious persecution has no boundaries and that it can afflict believers of every faith on every continent.
It will call on the Senate to censure "Senator Anning for his inflammatory and divisive comments seeking to attribute blame to victims of a horrific crime and to vilify people on the basis of religion, which do not reflect the opinions of the Australian Senate or the Australian people".
It will also call "on all Australians to stand against hate and to publicly, and always, condemn actions and comments designed to incite fear and distrust".
"In the face of attacks designed to sow division, our responses must bring us together, recognising an attack on any religion is an attack on all religions and that we all share a responsibility to unite, condemn and defeat such an attack on our common values and way of life," the motion states.
Mr Morrison, who attended a Coptic Christian church in Sydney on Sunday, said all Australians needed to stand together to show they were opposed to everything represented by the attack in Christchurch.
"We can all band together to stand against this hatred and this violence. We all stand together – people of many different faiths, or no faith at all – standing together for innocence and peace and for love over hate," he said.
"Love conquers hate and an attack on one faith is an attack on all faiths. This is the important message of hope today. In the midst of this awful atrocity, this is the hope that we can all cling to as peace-loving and free people all around the world."
Mr Shorten said "keyboard warriors" and right-wing politicians had to face up to their responsibility for the Christchurch murders.
"You by your hate speech have created a swampland of hatred through your ideology of hatred. You cannot disown what has crawled out of your swamp," he said after attending a Melbourne mosque.
"You can't disown your stupidity, your ignorance, your foolishness, your hatred, your racism, your intolerance, your un-Australian attitude and just hide behind hate speech."
Mr Shorten said the teenager who had hit Senator Anning with the egg had done a "mug thing" but that did not excuse the reaction of the senator's supporters.
"Right-wing extremist thugs. How many tough extreme right-wingers does it take to wrestle with a 17-year-old boy?" he said.
Victorian Chief Commissioner Graham Ashton said he saw the footage of the incident on Saturday night and instructed police to examine the video to determine whether any assault charges should be laid.
“I made it clear last night if there was any evidence that warranted assault charges then we’ll lay those charges,” he told reporters on Sunday.
Mr Ashton said police would examine everything, including the egging incident and the events that followed on from that.
"I think the entirety of that footage needs to be examined and we also understand we had federal police officers who may have been witnesses to that as well so we’ll talk to them," he said.
Greens leader Richard Di Natale said Senator Anning's actions were shameful and a disgrace to the Parliament, adding politicians and media outlets had "magnified" a message of hate.
One Nation NSW upper house candidate Mark Latham issued a statement saying: "Last week on social media Fraser Anning denounced our NSW One Nation campaign for being tolerant and opposed to religious discrimination.
"We wear that as a badge of honour. Anyone in Australian public life advocating for Nazi style 'Final Solution', as further reflected in his Christchurch comments, must be opposed at every opportunity."