The Ambulance Employees Association says the man died on Monday afternoon. He had been driving but pulled over and called triple zero when he experienced chest pains.
The association says a bystander started CPR but once paramedics arrived the man could not be revived.
Premier Peter Malinauskas said the death was "beyond tragic" and there would be a thorough investigation.
"A relatively young man has tragically lost their life under circumstances that may have been prevented," he told reporters on Tuesday.
"I understand from the information already on the record, this was a priority two case that was subsequently upgraded to a priority one.
"We need to see ambulances responding on time. It's not good enough."
The premier said his government was continuing to work to bring response times down during its first term in office but cautioned major changes would take time.
"That's a big undertaking because what we're seeking to do is turn around a very large ship that has been moving in the wrong direction for a long period of time," he said.
But he said the government was already making progress, adding 200 hospital beds, employing more ambulance crews, building more ambulance stations and recruiting more doctors and nurses.
"It's lots of separate efforts here that are going to be required her to turn this around."
The Ambulance Employees Association said at the time of the man's death there were 20 uncovered life-threatening emergencies across Adelaide with ambulances ramped at all major hospitals, some for more than six hours.
"This patient was only minutes away from hospital. Ramping deprived this patient of his best chance of survival," the association said.
