John Radford was left in a coma after the dispute in Huddersfield in 2013. He died 16 months later.
Michael Gledhill, who was found guilty of dangerous driving last October, was charged with causing death by dangerous driving after Mr Radford died.
Mr Radford was the chairman of Huddersfield and District Cyclists.
He had been involved in a campaign calling for better investigation of road accidents.
Gledhill, 24, of Bramble Bank, Holmfirth, appeared at Leeds Crown Court and was told he would be sentenced on 12 January.
The court had previously heard how words were exchanged between Mr Radford and Gledhill through the driver's side window as they travelled on the A616 on 31 July, 2013.
Contact was made between the car and bike, which resulted in Mr Radford falling into the road and suffering brain damage.
Speaking after Mr Radford was injured, his friend and fellow cyclist Martyn Bolt said: "There was nobody more experienced and knowledgeable about cycling.
"He was cycling in the right part of the road, just at the wrong time."
A motorist has admitted causing the death of a cycling safety campaigner who was knocked off his bike in a road rage incident.