He was involved in an accident while providing medical cover at a Skerries 100 practice session, the Motorcycle Union of Ireland said.
Dr Hinds, from Tandragee, was a consultant at Craigavon Hospital Area Hospital in County Armagh.
He regularly worked at the North West 200 motorbike races in County Antrim.
Dr Hinds, who lectured in trauma science, was taken to Beaumont Hospital in Dublin, where he died on Saturday.
Dr John Hinds and his colleague Dr Fred McSorley, known as the "flying doctors", have long enjoyed near legendary status among race fans.
The pair would follow racers, travelling very nearly as fast as many of them on their own high-powered bikes.
When riders crashed, invariably Dr Hinds would be there in seconds.
In 2009, he spoke to the BBC about his role.
"Hospital medicine has become very much full of protocols and guidelines - out here it's a wee bit more 'seat of your pants' medicine really," he said.
"And it's very rewarding to be able to treat people that way: where you don't have a sterile operating field and you're not in a resuscitation bay; you're very often in a ditch somewhere."
Dr Hinds had led a campaign for an air ambulance service to be introduced in Northern Ireland and had met with Health Minister Simon Hamilton to discuss the issue.
Mr Hamilton said Dr Hinds was a "lovely guy who will be sadly missed".
TUV MLA Jim Allister, who backed the air ambulance campaign, said the death of Dr Hinds was deeply shocking.
"He was one of our leading trauma experts. He was driven with a great passion to try and improve the chances of those involved in serious accidents. My thoughts are with his family.
"I do hope as a lasting tribute to him, the government will now push ahead with his ambition to have a proper air ambulance service for Northern Ireland.
"It would be very fitting for Dr Hinds," he said.
Members of the medical profession have also paid tribute to the Tandragee man.
The Northern Ireland Ambulance Service (NIAS) tweeted: "Saddened to hear reports of the untimely death of Dr John Hinds - a friend to many in NIAS."
Members of London's Air Ambulance service said Mr Hinds "was a true friend and advocate of the air ambulance community".
Dr John Hinds, one of the so-called "flying doctors" of Irish road racing, has been killed at a motorcycle event in the Republic of Ireland.