The 60-year-old man was watching the game in Edinburgh with friends when he collapsed about 15 minutes into the match.
The man was treated by paramedics and taken to Edinburgh Royal Infirmary, where he passed away a short time later.
“Inquiries are still ongoing but it is believed there are no suspicious circumstances,” Police Scotland said in a statement.
“We are extremely saddened to hear of the death of a supporter who attended today’s Calcutta Cup match,” Scottish Rugby Union president Donald MacLeod said. “We extend our deepest sympathy to the gentleman’s family and friends and we are standing by to support in any way we can.”