The accident happened on the A90 near Peterhead, Aberdeenshire, on Tuesday February 28.
It involved a Police Scotland Peugeot Partner van driven by a 46-year-old dog handler and a blue Vauxhall Astra.
The 58-year-old man driving the Astra and the police dog handler are now in a stable condition.
However, Sam, a nine-year-old black Labrador, was put down on Friday following complications after surgery.
Superintendent Steven Irvine, of the specialist operations unit, said: “As a force we care a great deal about our dogs and although it was an extremely difficult decision to make, it was agreed with very heavy hearts this was the best course of action for Sam.
“Sam was a much-loved and valued member of the Police Scotland family - as is the case with all our dogs - and he will be sorely missed.
“We are extremely proud of the fantastic crime-fighter Sam became and will always be grateful to him for the loyalty he showed his handler and the force.”