"We want police to arrest those who broke Baba Saheb's [Ambedkar's] statue. Otherwise, we will burn Maharashtra," a man shouted before being dragged away by police in Mumbai, India's industrial hub.
Ambedkar was born in Maharashtra and Mumbai is the capital.
Arup Pattnaik, Mumbai police joint commissioner, said: "There are reports of some sporadic incidents. However, things are under control."
But huge groups carrying banners were seen closing shops and clashing with police in several areas of the city.
Witnesses said police used batons and fired tear gas to disperse people in Mumbai's Worli, Bandra and Chembur neighbourhoods. Police denied the reports but said that several people had been arrested. Curfews were imposed in some towns.
Low-caste Hindus or Dalits, the name for those called "untouchables" in the past, make up about 16 per cent of India's 1.1 billion population, and are at the bottom of the 3,000-year-old Hindu caste hierarchy.
Though caste-based discrimination is banned, Dalits are still often beaten or killed if they use a well or worship at a temple reserved for upper castes.