Ryan, 39, was photographed being arrested at London City Airport last week in the aftermath of the incident.
Reports claimed that he had become disruptive after staff refused to serve him alcohol on the flight.
Apologising to fans on the band’s official Twitter account, the artist wrote: “A note from Lee. Over the last few days, I have had a lot of time to think about my behaviour and how it affects people around me.
“As it has been well documented, I have always had my struggles with alcohol and often say or do silly things as a result of that, which in turn has caused problems for myself and those around me. For that I want to apologise to everyone I have upset or offended.”
The singer admitted that he had “acted inappropriately on a flight to London” and disclosed that he was seeking professional help.
“I want to say sorry to the airline staff, everyone on the flight, as well as my band mates, my family and our fans,” he wrote.
“The time has come for me to take ownership of the root cause of these problems and for that reason, I am reducing my schedule for the time being to seek professional help and spend time with my family.
“I am aware of how my behaviour affects others and by starting this journey of working on my mental health I hope to emerge a better person for it.”
Reports suggested that Ryan spent over 12 hours in police custody, before being released pending further investigations.