The new law would further sharpen the communal divide and social polarisation and that would be detrimental to the unity and integrity of India, they said.
"To target Muslims, the CAA has provision to give citizenship to only those people who had come to India from three Muslim-majority countries Pakistan, Afghanistan and Bangladesh. Those from Nepal, Sri Lanka and Myanmar had been kept out of the law's purview with bad intentions," Bijan Dhar, Communist Party of India-Marxist (CPI-M) central committee member, said addressing a protest.
He said the Indian Constitution was always religion neutral, but the BJP government had amended it on religious lines to "establish a fundamentalist-dominated Hindu Rashtra."
Five Left activists, including a tribal leader Naresh Jamatia, and two policemen were injured in a BJP-CPM workers' clash at Belonia in south Tripura.
Five Left parties - CPI(M), CPI, CPI(ML)-Liberation, Forward Bloc and RSP - on Thursday called a nationwide agitation against the CAA.
Gautam Das, CPI-M Tripura secretary, said Narendra Modi-headed government had been implementing the "Hindu Rashtra" agenda of the Rashtriya Swaymsevak Sangh (RSS) since day one.
"The new citizenship law, along with BJP leaders' proclamations to extend the National Register of Citizens (NRC) to the entire country, was aimed to change India's character from a secular democratic one into a 'Hindutva Rashtra'," Das aid.
In the NRC's name, lakhs of people were being detained in detention camps in Assam, he said and added, similar camps were being set up in West Bengal, Maharashtra and other states.
( With inputs from IANS )