Bellew joined Ryanair in December 2017 after stepping down as chief executive of Malaysia Airlines as the Irish low-cost giant struggled with a staffing crisis.
The resignation comes as O'Leary prepares to become chief executive of Ryanair's new group structure - which will also include Austrian-registered Lauda and Polish-registered Buzz - and is due to hire a chief executive of the Ryanair airline.
Bellew "has done great work over the past 18 months to improve our operations, renew our engineering team, and significantly improve our operations management and on-time performance," O'Leary said in the memo.
Bellew was appointed after Ryanair announced the cancellation of around 20,000 flights after admitting it did not have enough standby pilots to operate a reliable schedule.
In a memo on Wednesday, Bellew told staff the company had a surplus of pilots and had frozen the hiring of captains, the Irish Times reported.
A spokesperson for Ryanair did not immediately respond to a request for comment.
Bellew was director of flight operations at Ryanair before he left in 2014.
He joined Malaysia as chief operations officer in September 2015 and took over as chief executive in July 2016 as it dealt with the aftermath of two tragedies in 2014, when flight MH370 disappeared in what remains a mystery and flight MH17 was shot down over eastern Ukraine.