"I would like to publicly apologise for drinking on that flight which is inappropriate behaviour," he said in a statement on the team's website.
"I know that I have let my team mates, coaches, management, sponsors and indeed my family down with my actions.
"Drinking on that plane was inexcusable and I know that I must take responsibility for it," added Henson.
Cardiff, who lost the Celtic League match 31-3 to Glasgow, said their management team would meet to discuss the matter further.
Henson, who missed last year's World Cup with a wrist injury and played no part in the 2012 Six Nations, joined the Blues in October after being released by French club Toulon.
He was interested in signing a new deal with the French Top 14 team but was suspended last April after unconfirmed media reports that he had fought with a team mate.
After the flight incident the 30-year-old said he needed to learn from it and make sure it never happened again.
"I admit I had been out drinking in Glasgow on the Friday night following our game against Glasgow Warriors and had stupidly carried on drinking on the flight for which I am truly embarrassed," added Henson.
"I can see that drinking and behaving inappropriately on that flight as a professional sportsman at 7am has caused offence to members of the public, the airline staff and passengers.
"I remain fully focused and committed to Cardiff Blues and I hope they can accept my apology." said Henson.
