Satadru Dutta, the event's chief organiser, has been detained by police, said Rajeev Kumar, director general of West Bengal police.
Messi was scheduled for a 45-minute visit to the stadium, but his appearance lasted just 20 minutes. Tickets for the event were priced from around 3,500 rupees ($38.65) - more than half of the average weekly income in India - but one fan said he had paid $130.
Fans in the Salt Lake stadium in the capital of the Eastern state of West Bengal threw ripped-up seats and other objects onto the field, while several people climbed a fence around the field and hurled objects.
"I can't believe there was so much mismanagement," said Eddie Lal Hmangaihzuala, who had travelled nearly 1,500 km from Mizoram over two days to attend the event.
"Messi left quickly, I think he felt unsafe. I hardly got a glimpse of him," he told Reuters.
The organisers of the tour did not immediately respond to a request for a comment.
"We've already detained the main organiser," Rajeev Kumar told reporters. "We're taking action so that this mismanagement does not go unpunished.
"He has already pledged in writing that tickets sold for the event should be refunded," he added.
Chief minister apologises, orders probe
Messi is in India as part of a tour during which he is scheduled to attend concerts, youth football clinics and a padel tournament, and to launch charitable initiatives at events in Kolkata, Hyderabad, Mumbai and New Delhi.
The chief minister of West Bengal, Mamata Banerjee, apologised to Messi and ordered a probe into the incident.
"I am deeply disturbed and shocked by the mismanagement witnessed today at Salt Lake stadium," Banerjee, who was on her way to the event when chaos broke out, posted on X.
"I sincerely apologise to Lionel Messi, as well as to all sports lovers and his fans, for the unfortunate incident," she said.
She said she was constituting a committee to conduct a detailed enquiry into the incident, assign responsibility, and recommend measures to prevent such occurrences in the future.
West Bengal, Kerala and Goa states have long had large football followings in otherwise cricket-crazed India.
Multiple incidents of fan riots have taken place at Salt Lake stadium, including during a Kolkata derby match between East Bengal and Mohun Bagan in 2012, which was abandoned after a brick thrown from the stands hit a player, seriously injuring him.
Argentine football great Diego Maradona twice visited Kolkata, and in 2017, unveiled a statue there of himself holding the World Cup in the presence of thousands of fans.
Messi, who captained Argentina for the first time in a friendly match at the Salt Lake stadium in a 1-0 win over Venezuela in 2011, virtually unveiled a 70-foot statue of himself in Kolkata earlier on Saturday.