The protests have been particularly vocal in the state of Assam, which recently saw four million residents left off a citizens' register.
The bill attempts to grant citizenship to immigrants who are not Muslim.
Students, activists, politicians and celebrities have all joined the protests against India's ruling party.
The Citizenship (Amendment) bill seeks to provide citizenship to non-Muslim migrants from Pakistan, Bangladesh and Afghanistan.
Supporters of the bill defend it by saying that Muslims have been excluded as the bill offers Indian nationality only to religious minorities fleeing persecution in neighbouring countries.
It comes months after the publication of the National Register of Citizens (NRC) - a list of people who can prove they came to the state by 24 March 1971, a day before neighbouring Bangladesh became an independent country. Around 3.62 million of those left off the register have submitted claims for inclusion again.
India said the process was needed to identify illegal Bangladeshi migrants.
Thousands of students have joined writers, artists and activists in regular protests against the bill, fearing that tens of thousands of Bengali Hindu migrants who were not included in the NRC may still get citizenship to stay on in the state.
Offices of the Bharatiya Janata Party (BJP), which runs both the federal government and Assam's state government, have been burnt down by angry mobs in many places.
Protesters have also frequently surrounded the state secretariat in the capital, Guwahati, demanding the shelving of the amendment.
Some supporters of Assamese peasant leader Akhil Gogoi even staged a nude protest in the national capital, Delhi, earlier this month.
"The movement against the bill has gained momentum across the north-east and if it is not withdrawn, the situation in the region may turn volatile," Sammujal Bhattacharyya of the All Assam Students Union (Aasu), which is leading the protests against the bill in Assam, told the BBC.
He said it was an attempt to provide citizenship "by backdoor" to illegal non-Muslim migrants who were excluded from the NRC.
In the early 1980s, Aasu was behind the anti-migrant protests in Assam that paralysed the state and degenerated into rioting that led to more than 3,000 deaths.
The agitation ended in 1985 after March 1971 was agreed as the cut-off date to determine citizenship.
India's main opposition Congress party opposes the bill on the grounds that determining citizenship on the basis of religion goes against the spirit of the constitution.
Regional parties who have joined hands with the BJP to form governments in Assam and the neighbouring states of Meghalaya, Nagaland and Mizoram have threatened to renege on their alliances because they all oppose the bill.
"Anyone who came to Assam after March 1971 is a foreigner, an illegal migrant. We don't care if he or she is Hindu or Muslim. Religion is not the issue here, it is a question of protecting indigenous people from being swamped by foreigners in their own land," university student Mitali Baruah told the BBC as she marched to a protest rally.
That has been the dominant sentiment with most of the indigenous communities - a migrant is unwelcome, regardless of religion.
"The BJP has failed to understand the pulse of the region, because they see everything through the prism of religion," said analyst Samir Purkayastha.
More about: India