Speaking in his first interview since his resignation, he told The Sunday Times: "I've never regretted being a priest.
"I've never felt unhappy, I've enjoyed it and tried to do whatever was asked of me.
"I've always gone where I've been sent and I hope to do the same again."
Mr Conry, 63, stepped down last weekend amid tabloid speculation of an affair with a parishioner.
In a statement read out at churches, he said: "I am sorry to confess that, going back some years, I have been unfaithful to my promises as a Catholic priest.
"I would like to reassure you that my actions were not illegal and did not involve minors.
"As a result, however, I have decided to offer my resignation as bishop with immediate effect and will now take some time to consider my future."
In the interview with The Sunday Times he goes on to to say that he doesn't want to defend his actions or call for a change in the laws of celibacy, but that he is glad his secret is out.
He said: "It's liberating, yes.
"Though I wouldn't want anyone to think I'm glad to be out of office."