A report described the body as "arrogant and controlling" and partly to blame for the incident in a Test match against South Africa in March.
Players Steve Smith, David Warner and Cameron Bancroft were banned.
Interim chair Earl Eddings said Cricket Australia needed to "earn back the trust of the cricket community".
The 145-page review was conducted by Sydney-based not-for-profit organisation The Ethics Centre.
Immediately following the review's publication on Monday, Peever, who had recently been re-elected for another three-year term, said he was still best placed to oversee the changes, but subsequently came under increasing pressure to resign.
"We look forward to continuing the important process of recovering and rebuilding for Cricket Australia and Australian cricket," said Eddings.
"The board is keenly aware that we have a way to go to earn back the trust of the cricket community. We and the executive team are determined to make cricket stronger.