The German suffered a 3-1 defeat at the Bernabeu, losing 9-2 on aggregate.
The coach told reporters: "Anyone who watched the game tonight saw that we restored our honour today. The team displayed a fighting mentality. Obviously Real had several chances to score, but [Ralf] Fahrmann showed a world-class performance.
"Two individual mistakes kept us from recording a draw here, but I was pleased with the display nonetheless. [Cristiano] Ronaldo didn't become the World Player of the Year for nothing. His qualities are outstanding."
Schalke have never progressed beyond the semi-finals in the Champions League.