Police in New Delhi said they have filed a complaint against Nupur Sharma and others over her remarks on the Prophet Mohammed and for "inciting people on divisive lines" on social media.
Crowds have protested in many areas across India over derogatory remarks about the Prophet Mohammad made by two former members of the ruling party.
Prime Minister Narendra Modi's government has faced a backlash from Muslims at home and abroad, including from a number of Gulf countries, after two officials from his Bharatiya Janata Party (BJP) commented in late May and earlier this month on the prophet's private life.
One of them, spokeswoman Nupur Sharma, has been suspended and the other official has been expelled by the party.
That has not calmed anger in the latest instance of what some in the minority Muslim community see as increased pressure under the rule of the Hindu nationalist BJP on issues ranging from freedom of worship to the wearing of hijab head scarves.
Tear gas fired at protesters
In the Uttar Pradesh city of Prayagraj, riot police with shields charged and fired tear gas at crowds on Friday in a street strewn with broken bricks and stones, video footage from Reuters partner ANI showed.
"Additional police forces have been deployed and strict action will be taken against those who are found responsible for this," said Prashant Kumar, a senior Uttar Pradesh police official.
In Jharkhand's capital Ranchi, protesters lobbed stones at police and injured some officers, senior police official Anis Gupta told Reuters by phone, adding the situation was now under control.
Protests in other cities and town remained peaceful.
Protests in Pakistan
In the Muslim-majority region of India adminstrated Kashmir, groups gathered in dozens of locations, some of them raising slogans against the fired BJP officials.
Authorities in the restive territory, which is also claimed by neighbouring Pakistan, cut off mobile internet connections to forestall any violence, a police official said.
In Ahmedabad, the main city in Modi's home state of Gujarat, protesters including children marched holding posters of the two BJP officials and calling for police to arrest Sharma.
"So far, no strict action has been taken against Nupur Sharma," said protester Mohammad Jabir. "She should be arrested as soon as possible."
Sharma was not available for comment. She said in a statement posted on Twitter last week that she did not intend to hurt anyone's religious feelings.
Police in New Delhi said on Thursday they have filed a complaint against Sharma and others for "inciting people on divisive lines" on social media.
The BJP has instructed officials to be "extremely cautious" when talking about religion on public platforms and has said it does not promote insults against any sect or religion.