Mr Laird was born in Greenock and has been involved in running companies in both the energy sector and sports management and events.
He took over as chairman at Morton in 2023.
A club spokesperson said: “Everyone at the club would like to thank John for his efforts and dedication in his time at the club, and wish him all the best for the future.
“Online and in-person abuse of any kind towards players, staff, officials or supporters is completely unacceptable and has no place in football or in society.”
Manager Dougie Imrie left the club in November to take over at Raith Rovers. The Cappielow side currently sit sixth in the Championship.
Former Motherwell and Nottingham Forest manager Billy Davies, who was appointed technical head coach at the club this summer, is currently leading a management team at the club as they search for a replacement for Imrie.