No, he didn't apologise on Twitter ('cause he quit, din 'eeh), or in a newspaper... he (and quite possibly Simon Cowell) decided it would be best to apologise on live TV, making Dermot O'Leary England's answer to Oprah.
Speaking on last night's X Factor, after performing of course, James Arthur said to ITV's lean mean hugging machine:
"I’ve made some silly mistakes. But I just want to thank all the people who've stuck with me, I wanna thank The X Factor for giving me the opportunity to do my dream job. Above all I wanna say sorry for abusing my position as an X Factor winner because I owe everything to this [show].”
And one can assume he was recently reminded of that by a certain high-trousered enthusiast.