The law fast-tracks citizenship for non-Muslim migrants from three neighbouring countries, but critics allege it is part of Indian Prime Minister Narendra Modi's Hindu nationalist agenda to marginalise the 200 million Indians who follow Islam.
In the country's northeast, however, even allowing non-Muslims citizenship is opposed by many locals who fear their culture is threatened by Bengali-speaking Hindus.
Modi, who insists he is not anti-Muslim, said the citizenship law is “1,000 per cent correct” and that Muslims from the three countries are not covered because they have no need of India's protection.
Rahul Gandhi, former opposition Congress chief, tweeted on Monday that the law and a mooted nationwide register of citizens also seen as anti-Muslim were “weapons of mass polarisation unleashed by fascists”.
On Sunday night in Delhi, police with batons fired tear gas and charged protesting students before storming a university.
On Monday fresh protests took place in Delhi, Chennai, Bangalore and Lucknow, where hundreds of students — most of them Muslims, television pictures indicated — tried to storm a police station, hurling volleys of stones at officers cowering behind a wall.
In the east in Kolkata, capital of West Bengal, thousands gathered for a major demonstration called by state premier Mamata Banerjee, a firebrand opponent of Modi.
In recent days empty trains were torched there and on Monday internet access remained suspended.
In Kerala in the south, another state whose government refuses to implement the citizenship law, several hundred people also protested.
Kerala's finance minister Thomas Isaac tweeted: “United action of all secular force is the need of the hour.”
Weekend of violence
Protests were reported in Mumbai, West Bengal, Aligarh, Hyderabad, Patna and Raipur over the weekend.
Authorities in northern Uttar Pradesh, meanwhile, have cut internet access in western parts of the state following demonstrations in Aligarh, home to a large university and a sizeable Muslim population.
The main epicentre of the protests has been in India's far-flung northeastern states, long a seething and violent melting pot of ethnic tensions.
There, where protesters are mostly Hindu, late last week four people died from gunshot wounds, one in a fire and a sixth beaten to death.
On Sunday night in Assam state — following days of rioting and clashes with police — around 6,000 people protested on Sunday evening, with no major incidents reported.
Modi blamed the main opposition Congress party and its allies of “stoking fire”, saying those creating violence “can be identified by their clothes” — a comment interpreted by some as referring to Muslims.
The UN human rights office said last week it was concerned the law “would appear to undermine the commitment to equality before the law enshrined in India's constitution”, while Washington and the European Union have also expressed concern.
The new law is being challenged in the Indian Supreme Court by rights groups and a Muslim political party, arguing that it is against the constitution and India's cherished secular traditions.
Ashok Swain, a professor at Sweden's Uppsala University said that the scale of the protests had caught Modi's government, which is presiding over a serious slowdown in economic growth, off guard.
“The protest is getting international attention and also spreading to different parts of the country. This certainly will add pressure on the regime when the economy has failed,” Swain told AFP.