Defender Nedum Onuoha's mother received a message sent accidentally by Cook which mocked her use of the phrase "ravaged with it" in the original email about her cancer treatment.
While Anthonia Onuoha received the reply, it was addressed to "Brian." City's football administrator Brian Marwood was also among the recipients of the woman's original email, which dealt with her son's contract.
Cook on Friday acknowledged an "error of judgment" and said his resignation was "in the best interests of the football club."
Although the email was sent in October 2010, Anthonia Onuoha only complained to City and English footballing authorities early this week.
The Sun newspaper, which received the email exchange from Onuoha, initially reported that Cook denied sending the email and said his email account had been hacked.
But City's board investigated her complaint and found the allegations were with "foundation," a club statement said.
"Garry Cook has offered his resignation, which has been accepted with regret by the board," City said on its website.
Chairman Khaldoon Al Mubarak said Cook "will always be welcome at Manchester City."
Cook was president of Nike's Jordan Brand until leaving the sportswear company after 19 years in 2008 to take the top job at Manchester City.
Within months, he oversaw the sale of the club by former Thai prime minister Thaksin Shinawatra to Abu Dhabi's Sheik Mansour, who has invested more than $1 billion to transform the team.
A 35-year trophy drought ended in May when City won the FA Cup while Roberto Mancini's side qualified for the Champions League for the first time.
"I am privileged to have held my position at Manchester City Football Club and to have experienced the opportunities that it has presented," Cook said on City's website. "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 judgment in this matter that has prompted me to reach this decision, which I believe is in the best interests of the football club."Suggest a correction