Kapoor, 67, the scion of a famed film industry family, is survived by his wife and two children and had been diagnosed with cancer in 2018.
“He remained jovial and determined to live to the fullest right through two years of treatment across two continents,” the family said in a statement.
The actor had moved to New York for treatment soon after the cancer was detected, returning to his hometown of Mumbai last September.
Kapoor’s death followed that on Wednesday of another Indian actor, Irrfan Khan, 54, who had roles in films such as “Life of Pi” and “Jurassic World”, and who also suffered from cancer.
Kapoor’s family urged his fans, who under normal circumstances would have been expected to pour onto the streets to mourn him, to follow novel coronavirus social-distancing rules.
“There are numerous restrictions around movement and gathering in public. We would like to request all his fans and well-wishers and friends of the family to please respect the laws that are in force,” the family said.
Kapoor came from a family of actors
His grandfather, Prithviraj Kapoor, father Raj Kapoor, brothers Randhir and Rajeev, and his son, nieces and nephews have all played roles in films.
He made his debut at the age of 16, playing a younger version of his father’s character in the 1970 film “Mera Naam Joker” (My Name is Joker), and later made a name for himself as a fresh-faced romantic hero.
Prime Minister Narendra Modi said he was anguished by Kapoor’s death.
“Multifaceted, endearing and lively … this was Rishi Kapoor Ji. He was a powerhouse of talent,” Modi said on Twitter.