Police say it was triggered by an argument between devotees at the shrine of Mata Vaishno Devi in Indian-administered Kashmir on Saturday.
Prime Minister Narendra Modi said he was "saddened" by the loss of life.
Tens of thousands of people usually visit the shrine every day, and it is thought there was a peak in visitors because of the holiday season.
Thousands of people had gathered on the first day of the new year at the cave shrine, which is among India's most revered pilgrimage sites, says the BBC's Anbarasan Ethirajan in Delhi.
The authorities had capped the number of visitors per day to 25,000 due to the pandemic.
But reports said almost twice the permitted number of devotees had gathered at the site, located near the town of Katra.
Another 16 people were injured in the crush which happened in the early hours of the morning.
After a brief pause, pilgrims have now been allowed to visit the shrine again, our correspondent says.
An investigation into the incident has been launched.
In a tweet, Mr Modi said: "Extremely saddened by the loss of lives due to a stampede at Mata Vaishno Devi Bhawan. May the injured recover soon."
Stampede at religious gatherings are not uncommon in India, when large crowds can gather with few safety measures.