The Portuguese was sacked on Tuesday after two and a half years in charge.
United are sixth in the Premier League, 19 points behind leaders Liverpool.
“I have been immensely proud to wear the badge of Manchester United since the first day I arrived, and I believe all United fans recognise this,” Mourinho said in a statement.
“Each time a chapter is closed, I show my deepest respect by not making any comments about my former colleagues.”
Former United striker Ole Gunnar Solskjaer was named the club’s caretaker manager until the end of the season on Wednesday.
Mourinho has gone back to his home in London, and the former Chelsea and Real Madrid boss has also asked the media to respect his privacy before he returns to football.
Speaking to Sky Sports News earlier on Wednesday, he said: “I was always very critical of managers that leave clubs and then they come out and they speak about the details of what happened, who is to blame.
“I could speak about the not-so-good, but that’s not me.
“Until I get back to football, I feel I have the right to live my normal life.
“Manchester United has a future without me and I have a future without Manchester United.”