Real parted company with Pellegrini last week after a disappointing campaign.
They lost out to Barcelona in the title race and were knocked of the Champions League by Lyon in the first knockout stage.
"I think a coach needs to be ready to be sacked," Mourinho said. "But if a coach fears that, he doesn't work well and has great problems. I am a coach that has a lot of self-esteem and confidence and I don't think about the possibility of being sacked.
"I think four years of contract is enough to win, to build a strong team for the present and the future. The sacking of Pellegrini doesn't make me happy. I am never happy when a coach is sacked.
"But football is like that.
"I have a lot of confidence in my players, I have the hope that my new players have confidence in me."
Mourinho shed little light on what changes he will bring to Real but outlined his vision for the future of the club.
He said: "It's too early to speak about changes. I believe it's time to analyse the situation, to get to know my club, ask lots of questions and get answers. I need to realise the direction of my work.
"The most important thing is not the coach or the players but the club.
"We are too small compared to the dimension of the club. If we work as a group, it's not difficult to get results. I want this group to have an identity, to win everything.
"The strength of my team is always the team and not the individuals.
"Cristiano Ronaldo is an incredible player not only for Real but for the whole of football.
"Cristiano is a winner and it's not difficult to persuade him of this."