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.
"Our family is so grateful for the overwhelming outpouring of love for our dear Tom."
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 cricket great 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.
After resolving to become an actor and attending a film institute in Pune, about 150 kilometres 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 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.
Prime Minister Narendra Modi, who offered condolences to Alter's family, recalled the actor's contribution to the film world and theatre.
"The world is a good man short this morning," Indian cricket commentator Harsha Bhogle tweeted. "Fine actor, pure sports lover and kind human being. You enriched our world."
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"While we mourn deeply, we are grateful he is now at peace," a statement from his family said.
"Our family is so grateful for the overwhelming outpouring of love for our dear Tom."
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 cricket great 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.
After resolving to become an actor and attending a film institute in Pune, about 150 kilometres 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 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.
Prime Minister Narendra Modi, who offered condolences to Alter's family, recalled the actor's contribution to the film world and theatre.
"The world is a good man short this morning," Indian cricket commentator Harsha Bhogle tweeted. "Fine actor, pure sports lover and kind human being. You enriched our world."
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