The chaos broke out as rumors spread that the bridge was collapsing over the Sindh River, said D.K. Arya, the deputy inspector general of police in the Chambal region of Madhya Pradesh state.
At least 64 people were killed, he said, while more than 100 were being treated in a hospital for injuries including broken bones.
Police wielding sticks had charged the crowd in an effort to contain the panic, Arya told the Press Trust of India. People retaliated by hurling stones at officers, and one officer was badly injured.
It was not immediately clear how many people were on the bridge when the stampede started. Local media said some 500,000 people had gone to the remote Ratangarh village temple in the Madhya Pradesh district of Datia to honor the Hindu mother goddess Durga on the last day of the popular 10-day Navaratra festival.
Sonia Gandhi, the leader of India's ruling Congress Party, expressed "shock and deep anguish over the tragic incident," according to a party statement.