Son of American missionary parents worked on more than 300 films, including both Hindi and international productions
Alter died at his home in Mumbai, India's financial hub and entertainment capital, late on Friday night.
He was diagnosed with skin cancer last year and had received treatment, but had suffered a relapse earlier this month.
Indian leaders as well as top film and theatre personalities expressed grief at his demise and offered condolences.
"While we mourn deeply, we are grateful he is now at peace," a statement from his family said.
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Born in India to US missionary parents, Alter spoke Hindi fluently and acted in both Bollywood and international films, in television shows and on the stage.
A sports journalist in the 1980s and 90s, Alter was the first person to interview Indian cricketer Sachin Tendulkar on TV, the Times of India said.
Alter was known for his role as a gangster on the 1990s soap opera Junoon, as well as for his work in Gandhi (1982) and One Night with the King (2006), the Hindustan Times reported.
In 2008, he received a Padma Shri award for distinguished service in art from the government of India.
Difficult decision
After resolving to become an actor and attending a film institute in Pune, about 150km outside Mumbai, Alter decided to renounce his US passport, according to a New York Times interview from 1989.
"You have to be truly committed to this country, otherwise you don't get respect or acceptability," Alter told the New York Times of the difficult decision.
Alter is survived by his wife, Carol Evans, son Jamie and daughter Afshaan. His body was cremated in Mumbai later on Saturday.
Narendra Modi, India's prime minister, offered condolences to Alter's family and recalled the actor's contribution to the film world and theatre.
