Their round of 16 tie was essentially settled when they were thrashed 6-1 at home in the first leg with Real also easing to a 3-1 victory over the Germans on Tuesday to book a last eight berth.
"The Champions League chapter is now unfortunately closed," said Schalke's Tim Hoogland, who scored their only goal in Madrid.
"But we have no reason to leave the competition with our heads hanging low. What is important now is to carve out a chance in the final league sprint so that we are back in the top European competition next season."
The top three German teams advance to the Champions League group stage with the fourth-placed team entering the qualifying rounds.
Schalke's inconsistent season indicated they would struggle on the tougher European stage but they have climbed up the Bundesliga table in recent weeks, moving into third past rivals Bayer Leverkusen thanks to last week's second straight win.
"Anyone who watched the game tonight saw that we restored our pride," coach Jens Keller said in Spain. "My team showed exactly that what we had wanted to show. We were aggressive and rehabilitated ourselves."
There was also a silver lining to the defeat with eight products of the club's youth system starting the game on Tuesday.
"It was exceptional to have eight players from the Schalke youth on the pitch at the start," said 19-year-old defender Kaan Ayhan, who was making his Champions League debut. "That is something that the entire club can be proud of."
Keller, however, will have to plan the immediate future without one of them when captain Benedikt Hoewedes suffered a suspected muscle tear and is expected to be sidelined for some time.
"We do not have a clear diagnosis yes but it is clear that it is a muscle injury," said coach Jens Keller. "He is not expected to play against Eintracht Braunschweig (on Saturday)."
- Sports & Recreation
- Tim Hoogland