Conaway played the swaggering teenager Kenickie, alongside John Travolta and Olivia Newton-John. He also played Bobby in the comedy series Taxi.
His family decided to switch off his life-support machine after he had spent several days in an induced coma.
The actor had battled for years against drug and alcohol addiction, something that he blamed on pain from a lingering back injury.
He spoke openly about his addiction on the Celebrity Rehab reality show in 2008, his last TV appearance.
Conaway made his Broadway debut in 1960 at the age of 10 in the Pulitzer Prize-winning drama All the Way Home.
He got his big break with Grease in 1978, but also suffered a back injury during filming which led to an addiction to painkillers and other drugs.
