The singer was caught on camera as he drunkenly relieved himself on a Central London ATM machine, just days after it was announced that the group had been chosen to perform at the Eurovision Song Contest in Dusseldorf.
"I was just so disgusted with myself when I saw those pictures," Costa told The Sun. "I was nearly in tears because I did think I'd blown it for the lads and myself.
"If the lads had said, 'We can't go on,' I would have been absolutely devastated and would probably have just gone away for many a year."
He continued: "All I can say is apologies. People who know me know I'm not like that. It was just so out of character."
His bandmate Duncan James commented: "We stick by each other. Ant was gutted the next day, but we didn't want to kick a man when he was down. He's our bandmate and you just move on from it. But he is never going to do it again. Lee was just glad it wasn't him!"
Blue will perform their track 'I Can' for the first time on The Graham Norton Show tonight on BBC One at 10:35pm.