The Portuguese head coach was officially unveiled by AC Milan on Wednesday, having spent the intervening period out of work following his dismissal after an 18-month spell at United that failed to deliver the turnaround the club had hoped for.
"The only thing I can say is that I learn a lot and I did some mistakes," said the 41-year-old. "I didn't have opportunity - and I am sorry for that - to say something to the Manchester United fans at the moment. I am really proud to have been their coach for a year."
Turning to his new role in Italy, Amorim struck a markedly more upbeat tone.
"It's an honour to be here. I'm so happy. I can feel that here people feel the club. I love Milanello, I think the place has history, it's quiet.
"People say that it is old. I love that kind of old, everything is there. I love the staff. I feel at home. I follow Milan since I'm young, so it's personal. It's hard to take this kind of challenge. The belief that I had at Sporting is the same belief I have now - with a little bit more hunger.
"Maybe because of the last adventure, I'm really hungry for success. I believe in myself and especially I believe in the club. I know it's a tough league, I know that you need to know Italian football really well, I think I learn fast and I will try to do that."
It is not the first time Amorim has reflected on his time in Manchester since taking the Milan job. Speaking separately to Calciomercato, he suggested his new challenge in Italy was, in some ways, even bigger than the one he faced at Manchester United.
"Why did I choose Milan? If you read an old interview of mine you will understand it, for me a special club," he said. "It's a very big challenge that I accepted even if after the last one I told myself to accept a smaller one. Perhaps it is a greater challenge than the last. I'm proud to be here, we all are, me and my staff."
Amorim replaces Massimiliano Allegri at San Siro and takes charge on a contract running until 2028, with United having since appointed Michael Carrick on a permanent basis in his old job.