Cocozza was axed from the programme on Tuesday amid claims that he boasted about taking cocaine. The X Factor has only confirmed that the singer broke a "golden rule".
Gary Barlow said that they couldn't discuss the incident for legal reasons, but Louis Walsh laughed: "I would say it was being unprofessional."
> Jonjo Kerr, Amelia Lily, 2 Shoes, James Michael: New X Factor pictures
Walsh also brushed off criticism from Barlow for his comments about Cocozza on last Saturday's show, claiming: "I'm sorry guy, I was just being honest."
All four X Factor judges were speaking at a conference to introduce their four returning acts - Amelia Lily, Jonjo Kerr, James Michael and 2 Shoes - who will compete in a public phone vote for the right to replace Cocozza in the talent show.
> 'X Factor' phone vote to close during Saturday's show
Scroll through pictures from today's X Factor press conference below: