The Portugal forward has deflected attention away from his embattled compatriot with outstanding displays, scoring twice against Real Sociedad in La Liga on Sunday and a hat-trick in the King's Cup against Celta Vigo on Wednesday.
Real's top scorer, with 174 goals from 172 appearances and the runner up to Lionel Messi in Monday's World Player of the Year awards, is, however, suspended for their next game away at Osasuna in La Liga on Saturday.
"I have to ask that they [the fans] stop this now," Ronaldo told reporters after their 5-2 aggregate victory set up a cup quarter-final with Valencia.
"The boss is going to stay, we still have a lot we can win this year. He is in charge and he takes the decisions, the fans and the players should help him.
"People have shown they are unhappy but that's enough now. They have to get behind the boss and the team. It affects the players out on the pitch. If we win the Cup and the Champions League it will have been a good year."
The inconsistent performances combined with Mourinho's confrontational style, hints that he may leave, and the recent decision to drop captain and fan-favourite Iker Casillas have alienated sections of Madrid's more traditional supporter base.
The last two home games have resembled plebiscites with huge cheers for Casillas when his name was read out on the team-sheet before kick-off, while Mourinho's has been met with a mixture of jeers, whistles and applause.
"I have felt the support of the Bernabeu for some time now and it makes me happy," Ronaldo added. "If the fans are with me, it motivates me."
"I have a contract and I am happy here but as you say, anything is possible in the world of football," he said.
Real have only won once in their last four league visits to Osasuna's compact stadium in Pamplona, but cannot afford any more slip ups as they try to cut the gap on the unbeaten leaders Barca, who have to visit fourth-placed Malaga on Sunday.
