Casillas, who is also captain of world and European champions Spain, was left on the bench for the second match in a row, after also missing the 3-2 defeat at Malaga, and the decision did not go down well with home fans at the Bernabeu.
Casillas, a big fan favourite, was cheered by supporters when his name was read out among the substitutes before kickoff, but Mourinho's name was met with whistles from sections of the crowd.
"If they whistle me for leaving Iker on the bench, fine, and if they whistle me for poor performances, that's OK by me," Mourinho told a post-match news conference when asked about the crowd's reaction.
"I don't want to be a hero all the time, independent of my performances."
Adan was only on the pitch six minutes before being sent off for bringing down Carlos Vela in the area, and Casillas entered in his place to a huge ovation from the Bernabeu, though he failed to stop the ensuing penalty.
A double from Cristiano Ronaldo helped Real go on to win the game.
"The fans and the media don't have a problem with Adan, they have a problem with me for my decision, but Adan suffers for this pressure," Mourinho added.
"I think it shows a lack of respect towards Adan. He has been at Madrid 16 years, he came through the youth ranks and he has the right to be happy. You [the media] have turned a problem with Mourinho into a lack of respect towards Adan."
Mourinho's controversial decision to drop Casillas, the first name on the Real team-sheet for more than a decade, has been a hot topic of conversation over the mid-season break.
He defended his original decision as being based on 'technical reasons' in Malaga.
In Saturday's pre-match news conference the former Chelsea and Inter Milan boss spoke of the necessity for competition for places.
Adan pursued a similar line to Mourinho.
"I know I have achieved the position because of my hard work," the 25-year-old told reporters. "Many of the opinions have shown a lack or respect towards me. I am another member of this squad.
"The boss picked me for this game and it was my intention to do my best. Whatever people say, the boss and I will not take much notice.
"I was gutted when I returned to the dressing room. The red card seemed excessive. It's a penalty but it was unfair to send me off."
With Adan now suspended for next week's league game at Osasuna, it remains to be seen who will don the gloves for Wednesday's visit of Celta Vigo in the King's Cup.
Real need to overcome a 2-1 deficit at home to reach the quarter-finals, and Mourinho said on Saturday whoever played against Sociedad would not play in the cup.
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