Seeking to soothe the frayed nerves, senior Union Ministers Rajnath Singh and Ananth Kumar said the individual has already apologised and the country is secular.
With noisy protests by Opposition members who also demanded lodging of an FIR against him on the issue, the House proceedings were adjourned thrice -- once during the Question Hour and twice during the Zero Hour.
Attacking the ruling party over the remarks, Congress leader Mallikarjun Kharge wondered whether the people from South India were not Indian citizens.
Wanting to know what action would be taken against Vijay who is "not an ordinary person", the Congress leader said he was a former Rajya Sabha member and has also written many books on BJP philosophy.
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His remarks are a "threat" to the unity of the country and "you want to divide this country", Kharge said, adding that if such things go on, then states would start asserting for independence.
This "shows your mentality", he said as he sought to compare them to the likes of Hitler and condemned the remarks. "FIR should be lodged against him (Tarun Vijay). He has talked about breaking the nation. It is anti-national," Kharge said.
Since the individual has already apologised, there is no need to raise any more questions about it, the Home Minister said, adding he would not mention the name.
As the protests continued even after Singh's statement, Speaker Sumitra Mahajan said "this (House) is not a court."
Parliamentary Affairs Minister Ananth Kumar sought to soothe the agitating members saying, "We are all one. The country is one".
BJP is of the view that "we all are Indians. There is no difference on the basis of colour. We all are one. Congress should not create differences. We are all together".
"Khargeji please don't try to create differences... One India. Great India," he emphasised.
Earlier during Question Hour, Kumar urged the Opposition to allow Commerce Minister Nirmala Sitharaman to respond to the question posed to her in Question Hour, saying she was a minister "from the South".
During Question Hour also, the Opposition members raised slogans in the Well.
The Opposition members were heard shouting, 'Modi sarkar jawab do-jawab do (Modi government should respond)', 'Desh ka vibhajan nahin chalega (Won't allow division of country)' and 'We want FIR (against Vijay)'.
With the din continuing during the Zero Hour, the Speaker adjourned the proceedings for over 20 minutes till 1245 hours.
The House was again adjourned till 1350 hours as the protests continued.
There was also a brief adjournment of the House over the issue for 10 minutes till 1120 hours during the Question Hour.