The Edinburgh-based company said Jeremy Stafford had tendered his resignation "for personal reasons".
Mr Stafford joined Menzies in October 2014, having previously been chief executive of Serco UK and Europe.
His departure comes a few weeks after Menzies issued a profits warning following problems with its ground handling contract at London Gatwick.
In a trading update on Wednesday, Menzies said negotiations over the airport contract had been "satisfactorily concluded".
It added that since its previous update in November, trading had continued to be in line with its forecasts and it remained confident of meeting expectations for the full year ended 31 December 2015.
The group also announced that former Menzies Distribution managing director Forsyth Black had been appointed as managing director of Menzies Aviation with immediate effect.
Forsyth has been with the group for more than 16 years, predominantly in senior aviation roles.
Mark Cassie, currently Distribution Operations Director, has been appointed as interim boss of Menzies Distribution.
John Menzies chairman lain Napier said: "I am delighted to welcome Forsyth to the board with his extensive and relevant experience which we are confident will drive forward the implementation of our existing aviation strategy.
"I also look forward to working closely with Mark Cassie as he continues to develop our distribution business into the e-commerce fulfilment market.
"The board wish Jeremy Stafford well for the future."
The chief executive of logistics group John Menzies has resigned after just 15 months in the post.