He said: "It was a very, very difficult decision for us to make, but it was right for myself and for the football club.
"I have had fantastic support from the board from minute one right up until the time of leaving, but we are in a results-driven business and I don't like being second best at anything."
Chairman Roger Munby added: "We had a long discussion with Peter after the QPR game in which Peter said that he felt he had been supported by the club in everything that he had tried to do at the Carrow Road.
"Therefore he felt he owed it to the board to be honest with himself and with us about whether he could turn things round."
Jim Duffy has been placed in temporary charge of first-team affairs with the Canaries following Grant's exit.