Six people died during clashes between demonstrators and police in the northern state of Uttar Pradesh on Friday, taking the nationwide death toll to 14.
After thousands took to the streets of Uttar Pradesh, police imposed a British colonial-era law banning the assembly of more than four people is some parts of the state.
In the northeastern border state of Assam, where internet services were restored after a 10-day blockade, hundreds of women staged a sit-in against the law in Gauhati, the state capital.
"Our peaceful protests would continue till this illegal and unconstitutional citizenship law amendment is scrapped," said Samujjal Bhattacharya, the leader of the All Assam Students Union, which organized the rally.
He rejected an offer for dialogue by Assam Chief Minister Sarbananda Sonowal, saying talks cannot take place when the "government was hoping to strike some compromise."
In New Delhi, police said they arrested 15 people in connection with the late Friday night violence in the Daryaganj area during a protest. Those arrested were charged with rioting and using force against police.
The government also issued an advisory asking news channels to refrain from broadcasting content "likely to instigate violence."
India's Ministry of Information and Broadcasting, which released the advisory, asked for its "strict compliance." The ongoing backlash against the law marks the strongest show of dissent against the Hindu nationalist government of Prime Minister Narendra Modi since he was first elected in 2014. Many of the protesters are angered by the new law that allows Hindus, Christians and other religious minorities who are in India illegally to become citizens if they can show they were persecuted because of their religion in Muslim-majority Bangladesh, Pakistan and Afghanistan. The law does not apply to Muslims.
Critics have slammed it as a violation of the country's secular constitution and label it the latest effort by the Modi government to marginalize India's 200 million Muslims. Modi has defended the law as a humanitarian gesture.
Malaysian Prime Minister Mahathir Mohamad on Saturday criticized the law as unfair.
At a news conference following the conclusion of an Islamic summit in Kuala Lumpur, Mahathir said India is a secular state and the religions of people should not prevent them from attaining citizenship.
"To exclude Muslims from becoming citizens, even by due process, I think, is unfair," he said.
Protests against the law come amid an ongoing crackdown in Muslim-majority Kashmir, the restive Himalayan region stripped of its semi-autonomous status and demoted from a state into a federal territory last summer.
They also follow a contentious process in Assam meant to weed out foreigners living in the country illegally. Nearly 2 million people were excluded from an official list of citizens, about half Hindu and half Muslim, and have been asked to prove their citizenship or else be considered foreign.
India is also building a detention center for some of the tens of thousands of people the courts are expected to ultimately determine have entered illegally. Modi's interior minister, Amit Shah, has pledged to roll out the process nationwide.
Critics have said the process is a thinly veiled plot to deport millions of Muslims.