Spain defender Ramos, who was selected in FIFA's team of the year on Monday, picked up a second yellow card when he brought down Augusto Fernandez in the 73rd minute in Wednesday's 4-0 last-16 second leg victory at the Bernabeu.
The 26-year-old directed insults at the referee and his assistant when leaving the pitch, according to the referee's report, and was punished with one game for the red card and four for his comments to the match officials.
"Independently of whether I think it is fair or not, I want to apologise to the coaching staff, my colleagues and the fans," Ramos tweeted.
"I hope they will be as equally 'strict and fair' with the referees that make mistakes game after game."
Real have the right to appeal the decision which, as it stands, rules Ramos out of the La Liga games against Osasuna, Valencia and Getafe, and both legs of the Copa quarter-final against Valencia.
The RFEF have taken a tough stance on insults to referees this season.
The loss of Ramos is a blow to coach Jose Mourinho, who is already without his other first-choice centre-back Pepe.
The Portugal defender had surgery on his ankle last week and could be out for around a month.
Full-backs Marcelo and Fabio Coentrao have just returned to training after injury but have yet to play in 2013.