Students pelted stones at police who locked up the gates of a college in the northern city of Lucknow to prevent them from taking to the streets. About two dozen students at another college in the city sneaked out to protest.
Anger with Prime Minister Narendra Modi's government was stoked by allegations of police brutality at Jamia Millia Islamia university on Sunday, when officers entered the campus in the capital New Delhi and fired tear gas to break up a protest. At least 100 people were wounded.
Critics say the law, which does not make the same provision for Muslims, weakens India's secular foundations.
Cars were attacked in Delhi on Sunday and public buses set on fire.
The head of Jamia Millia demanded an investigation into how police were allowed to enter the campus. "It is not expected of the police to enter the university and beat up students," Najma Akhtar told a news conference.
Students said police fired tear gas and windows were broken in the library. They ducked under desks and switched off the lights as advised by teachers. Police said they acted with restraint.
Rahul Gandhi, leader of the main opposition Congress party, said the Modi government was dividing Indian society through the citizenship law and a plan to launch a national citizenship register.
"The best defence against these dirty weapons is peaceful, non-violent Satyagraha," he said in a tweet referring to the strategy of passive political resistance advocated by independence leader Mahatma Gandhi.
Rahul Gandhi's sister Priyanka Gandhi sat down for a protest at the India Gate war memorial in Delhi against the police crackdown on student campuses.
Students at Delhi University said right-wing student groups linked to Modi's party were trying to intimidate them for protesting against the citizenship law.
Sanija Mithran, another student of the college, said girls who were wearing the hijab were targeted. "They were selecting the girls with hijab and they were pulling their hijab."
Both steps are seen advancing a Hindu-first agenda.
Modi's Bharatiya Janata Party denies any religious bias. It says the new law is meant to help minority groups facing persecution in the three nearby Muslim countries.
The most violent protests during the past few days occurred in the northeastern state of Assam, where mobs torched buildings and train stations, angry the law would help thousands of immigrants from Bangladesh become lawful citizens. At least two people were killed.