The former Monza goalkeeper left for Juve last week.
Di Gregorio said: “I feel like I did on the first day. Every day feels like the first day. Everything is really great. Then I got to see the facilities, and everything is really great.
“When the season was over, I spoke with director (Cristiano) Giuntoli and I didn’t think about any other teams anymore. I gave my word, and I kept it.
“I think, as you said, I made a path from the bottom. It helped me to play, to play a lot. Making mistakes, making mistakes, working on them. I’ll tell you, the work, the perseverance, has led me to be here today.”
He added, “Yes, history certainly matters, even for goalkeepers. Juventus has had world-class goalkeepers. For me it’s great to be here. As I said, I also feel very proud for this reason.”