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Police used water cannon to disperse protesters when they tried to march on the office of Chief Minister Akhilesh Yadav in Lucknow, capital of the northern state.
Yadav is from the regional Samajwadi Party that suffered heavy losses to Modi's Bharatiya Janata Party (BJP) in the recent general election. He come under fire over lawlessness in his state and the failure of police to investigate sex crimes.
State governments are responsible for law enforcement in India. The BJP protesters, mainly women, also called for the imposition of direct presidential rule in Uttar Pradesh, India's most populous state.
No BJP leader has visited the village where the two girls, aged 12 and 14, were murdered last Tuesday. Modi has yet to comment on the killings.
A cabinet minister, from a party allied to the BJP, became the first high-ranking representative of Modi's week-old government to visit the scene on Monday.
"The whole country is shaken but neither the chief minister nor any minister who are there has come to this place," said Ram Vilas Paswan, who is Modi's minister for consumer affairs, food and public distribution.
"This means that either they are scared of the public or they are trying to protect the culprits," he added.
Three suspects who are brothers from the powerful Yadav caste have been arrested along with two police constables.
Akhilesh Yadav, the chief minister, has since called for the murders to be investigated by India's top crime agency, the Central Bureau of Investigation. Officials from his party denied that the government had failed to act over the killings.
"In the Budaun rape case, I feel that every possible measure that could have been taken has been taken," Gaurav Bhatia, a Samajwadi party spokesman, told news channel CNN-IBN.