Cook stepped down from his post with the Premier League club following an internal investigation into the circumstances surrounding an email sent to Dr Anthonia Onuoha, the agent and mother of City player Nedum Onuoha.
Cook had denied that he sent the email, claiming someone hacked into his email account.
However Cook's resignation was confirmed on Friday following an internal investigation by City.
"The club can confirm that there is a foundation to Dr Onuoha's allegations and the chairman has written to apologise to Dr Onuoha for any distress caused," a statement said.
"Following the findings of the review, Chief Executive Officer Garry Cook has offered his resignation, which has been accepted with regret by the Board."
According to reports, Dr Onuoha wrote to City official Brian Marwood about a contract dispute between the club and her son, who is an England Under-21 international.
In an email sent to Marwood last October, she said that, although her body was "ravaged with cancer and ongoing chemotherapy", she was still able to fight her son's corner.
Two weeks later she received an email, apparently written by Cook but destined for Marwood and sent to Dr Onuoha by mistake.
The Sun claimed: "The e-mail was addressed 'Brian' then said: '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 (sic).'"
A former executive at sportswear giant Nike, Cook was the subject of negative headlines in the early days of his tenure at City.
He once described controversial former City owner Thaksin Shinawatra as "a great guy to play golf with," and said an attempt to sign Brazilian superstar Kaka failed because the player "bottled it."
Source: AFP
Source: AFP