Speaking in the stadium he said: "I never imagined that fans would have such a feeling for me.
"It was impressive. A special moment for me today."
80,000 people crowded into Real Madrid's Santiago Bernabeu Stadium
He also talked about his time at Manchester United, saying: "I have to say thank you to all supporters, my boss, my friends and my team-mates because I'd been there six years and I had a good relationship with all people there.
He'll miss Manchester
"But the people understand my decision and the people respect that. Of course it's a great challenge now and the people know my dream is to play in Madrid.
"I'm going to miss many many things in Manchester but this is the life, this is my new house, my new club and I'm going to give 100 per cent for this club."