Morata was booed by fans due to his Atletico Madrid links - despite also being a former Real Madrid player.
De la Fuente later said: "It hurts my soul that in my country they whistle at the captain of the National Team and when I hear those whistles I feel pain.
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"In the end there are always people who bring you down. The Bernabu has been enormous, but as a Spaniard I feel ashamed when a player from Spain is whistled.
"But the vast majority has supported Spain. It is everyone's job to educate in support of the National Team. Here we have to leave aside the colors of the clubs, and that is a job for the press.
"Let's see if we can all solve this."