Cook resigned from his position at the Etihad Stadium after an internal investigation concluded "there is foundation" in allegations he sent an offensive e-mail to a player's mother.
Cook appeared to make light of the condition of cancer sufferer Dr Anthonia Onuoha, mother of defender Nedum Onuoha, in a message intended for the club's football administrator Brian Marwood.
Dr Onuoha, representing her son during a contract dispute, had emailed Cook saying that while she was "ravaged with cancer and ongoing chemotherapy" her mental faculties were still intact.
The reply she received, apparently by mistake, was addressed to 'Brian', and read: "Ravaged with it! I don't now how you sleep at night. You used to be such a nice man when I worked with you at Nike."
Cook denied he sent the message and claimed his e-mail had been hacked, possibly by a rogue employee.
However, after further evidence emerged suggesting the email had originated from Cook's iPad, the board looked into the matter.
A City statement read: "The club can confirm that there is foundation to Dr Onuoha's allegations and the chairman has written to apologise to Dr Onuoha for any distress caused.
"Following the findings of the review, chief executive officer Garry Cook has offered his resignation, which has been accepted with regret by the board."
Cook, whose tenure was blighted by several public relations gaffes, also expressed regret.
He said: "I am privileged to have held my position at Manchester City Football Club and to have experienced the opportunities that it has presented.
"The privilege is in part offset however by the significant personal focus which has at times, detracted from the magnificent achievements of those working at the football club.
"It is that factor, together with my error of judgement in this matter, that has prompted me to reach this decision, which I believe is in the best interests of the football club."
Cook's previous blunders included referring to City's previous owner Thaksin Shinawatra, who was found guilty of corruption in a Thai court, as a "nice guy", criticising former player of the year Richard Dunne and accusing AC Milan of "bottling it" over the possible sale of Brazil playmaker Kaka to City.
Despite that, Cook did make a positive impact during his time at City, overseeing the club's rise to become major players on the European stage.
Chairman Khaldoon Al Mubarak said: "Garry has made a remarkable contribution to Manchester City Football Club over the past three years.
"His judgement in this matter should in no way lead to his accomplishments being overlooked."