As chaos broke out, a 39-year-old woman and her nine-year-old son were pulled out of the crowd as they felt "suffocated", police said.
They were immediately given emergency first aid, before being taken to hospital for further care.
While the woman died at the hospital, her son was shifted to a different hospital where he is still undergoing treatment.
The actor has not yet commented on the police's allegations.
But shortly after the incident, he took to X, formerly Twitter, saying that he was "heartbroken by the tragic incident".
"My heartfelt condolences go out to the grieving family during this unimaginably difficult time. I want to assure them they are not alone in this pain and will meet the family personally," he wrote.
He later announced assistance of 2.5m rupees ($29,480; £23,346) for the woman's family and promised to take care of the medical expenses for her son.
Mythri Movie Makers, the studio behind the film, also released a statement saying, "We are committed to standing by them and extending all possible support during this difficult time."
Arjun had approached the court earlier this week to quash the police complaint against him but the case hasn't been heard yet.