Heather Cho, whose Korean name is Cho Hyun-ah, resigned from the airline Tuesday, the airline said.
As for the next steps, the spokesman told CNN there is an investigation going on, so "we will have to see."
The incident occurred Friday at New York's JFK airport on a flight due to take off for the South Korean city of Incheon.
Cho reportedly demanded that the plane go back to the gate so the crew member who served her macadamia nuts out of a bag instead of on a plate in First Class could be kicked off the flight.
Cho's job was to manage in-flight service, but she was only a passenger on this flight and was not flying in an official capacity.
According to South Korean news agency Yonhap, Cho is a vice president of the national carrier and the eldest daughter of Korean Air's chairman, Cho Yang-ho.
He apologized Tuesday for any inconvenience caused to passengers and said the incident would be investigated, the news agency reported.
Korean Air also apologized for any discomfort to those on the flight and said there had been no safety issues involved. The plane arrived at its destination only 11 minutes behind schedule, according to Yonhap.
"Even though it was not an emergency situation, backing up the plane to order an employee to deplane was an excessive act," the airline said in a statement.
"We will reeducate all our employees to make sure service within the plane meets high standards."
By Ruchi Singh