The 67-year-old actor was admitted to a hospital in Mumbai on Wednesday.
He had received cancer treatment in the United States for a year and came back to India in September last year.
Born in a prominent Bollywood dynasty that has produced many stars, the late actor made his mark on the industry with the 1973 romantic film "Bobby".
Rishi's well-known movies include "Karz", "Nagina", "Saagar", "Amar Akbar Anthony" and "Chandni".
He began acting with a role in his father Raj Kapoor's famous 1970 movie "Mera Naam Joker".
Indian Prime Minister Narendra Modi described Rishi as "a powerhouse of talent" as condolences poured in from the film industry and politicians.
"We have lost one of the greats today. An amazing actor, a wonderful human being, and 100 percent a child of cinema," film star Aamir Khan said.
Rishi is survived by actress wife Neetu Kapoor, son Ranbir and daughter Riddhima.
In his death, India's film industry has lost second top actor in two days.
Irrfan Khan, widely known for his performances in "Life of Pi" and "Slumdog Millionaire", died on Wednesday at a hospital in Mumbai.