Muslims took to the streets after Friday prayers in huge numbers across India and neighbouring countries to condemn the remarks, with police firing on a crowd in the eastern city of Ranchi.
"Police were forced to open fire to disperse protesters... resulting in the death of two," a police officer from the eastern city of Ranchi told AFP.
Officers said that the crowd had defied their orders not to march from a mosque to a market and thrown broken bottles and stones when police attempted to disperse the rally with a baton charge.
Authorities cut internet connections in the city and imposed a curfew, with local resident Shabnam Ara telling AFP the atmosphere remained tense on Saturday.
"We are praying for peace and harmony," she added.
Police in Uttar Pradesh fired tear gas to disperse at least one rally after several demonstrations were staged across the northern state.
Most protests ended peacefully but demonstrators in some cities had thrown stones at police and injured at least one officer, said Avanish Awasthi, a senior government secretary in the state.
"We will take strict action against those indulging in stone pelting and violence," Awasthi told reporters.
"Those working behind the scene, instigating violence, will not be spared at all."
Prashant Kumar, a senior police officer in the state, told AFP that up to "136 protesting miscreants" had been arrested from six districts around Uttar Pradesh.
Cities around India saw sizable demonstrations on Friday, with some crowds burning effigies of Nupur Sharma -- the Bharatiya Janata Party (BJP) spokeswoman whose comments set off the furore.
Sharma's remarks have embroiled India in a diplomatic storm, with the governments of nearly 20 countries calling in their Indian envoys for an explanation.
The BJP has gone into damage control, suspending the official from its ranks and issuing a statement in which the party insisted it respected all religions.
Friday saw the biggest street rallies yet in response to the furore, with police estimating more than 100,000 people mobilised across Bangladesh after midday prayers.
Another 5,000 people took to the streets in the Pakistani city of Lahore at the behest of a radical religious party, calling on their government to take stronger action against India over the comments.
The row follows anger across the Muslim world in 2020 after French President Emmanuel Macron defended the right of a satirical magazine to publish caricatures of the prophet.
French teacher Samuel Paty was beheaded in October 2020 by a Chechen refugee after showing the cartoons to his class in a lesson on free speech. Images of the prophet are strictly forbidden in Sunni Islam.