Assistant Chief Constable Mark Milsom – a member of the force’s command team – has admitted responsibility for the collision, which happened on New Briggate in June.
After a long internal investigation, Chief Constable Mark Gilmore confirmed a traffic offence was committed.
But he said it was not in the public interest to bring charges.
“Assistant Chief Constable Milsom had accepted responsibility from the outset,” he said.
“The investigation itself found that a road traffic offence occurred and supported the view that the driver of the bus was not to blame, recommending that the normal decision-making process applied to ordinary members of the public should also apply in this case. Following careful consideration, I decided that it was not in the public interest to prosecute.”
The incident happened on June 5 as Mr Milsom – who will take charge of policing the Yorkshire element of next year’s Tour de France – drove to work in his BMW X5. No-one was injured.
The West Yorkshire force’s investigation was independently reviewed by South Yorkshire Police.
Mr Gilmore said Mr Milsom had been on a driver training improvement course, run by the force, following the incident.
The Alliance for British Drivers, which campaigns against speed cameras and fuel taxes, criticised the decision not to prosecute.
Founder member Hugh Bladon said: “I don’t see why being a police officer allows him to escape any form of normal punishment. The only time a police officer should be able to get away with this sort of thing is when he’s on an emergency call.
“They should have thrown the book at him.”