A spokeswoman from Police Scotland confirmed that an elderly man had died after taking unwell at the game at 3.10pm Saturday afternoon.
They said there were no suspicious circumstances surrounding his death and a report would be sent to the procurator fiscal.
With only 10 minutes played, referee Don Robertson halted proceedings after becoming aware of a problem in the crowd before taking the players from the field.
Medics attended to the supporter in the stands and after the serious nature of the incident became more apparent, the decision was taken to call off the match by both teams as a mark of respect.
A spokesman from the Scottish Ambulance Service confirmed an ambulance was in attendance at the scene at Cappielow in Greenock.
A tweet from Greenock Morton's twitter account read: "Due to a tragic incident in the crowd, both clubs have agreed to abandon the match. A date for a replay will be announced in due course."
The club later released a statement saying: "It was with deep regret that our match today against Queen of the South F.C. was abandoned due to the tragic death of one of our loyal supporters during the game.
All at Greenock Morton Football Club would like to take this opportunity to extend our heartfelt sympathy to all family and friends of the gentleman concerned.
We would further like to register our appreciation for the efforts of the emergency services, St Andrews First Aiders and Stewards who carried out their duties in a professional and caring manner throughout.
And finally, our thanks must also go to supporters of both teams who showed patience and great respect as events unfolded."