The Northern Ireland international signed a three-year contract with Gordon Strachan's side earlier this week and was unveiled at Celtic Park on Thursday morning.
McCourt said: "It's fantastic. It's something that I've always dreamt about my whole life and to get the opportunity to come play for this club was unbelievable.
"It was a shock but a good shock.
"To be honest, I was nervous when I heard that Celtic's interest was serious.
"I wanted the medical and everything to go okay and it was a relief to sign.
"It's only in the last few days that I've been able to enjoy it and look forward to coming in next week.
"I have been here at Celtic Park as a fan before but it was fantastic to have a look again and take it all in.
"I'm not looking much further than pre-season and then I'm going to take it from there."