Spain lifted the trophy with a 1-0 extra-time victory over Argentina in New York on Sunday, but the celebrations were overshadowed by confrontations after the final whistle.
Argentina players, including Atletico Madrid defender Nahuel Molina and midfielder Leandro Paredes, were among those involved in the incidents.
De la Fuente admitted he did not see what happened immediately because he was celebrating with his players, but said the scenes he later witnessed were unacceptable.
"At the time, I didn't realise it. I was celebrating, hugging teammates," De la Fuente told Spanish television.
"But in any case, it is intolerable, unacceptable, it cannot be allowed. Players of that stature, whom I praised in the days leading up to the match and continue to praise, who are great players with a tremendous coach, surely they must have felt just as bad as we did seeing those reactions."
The Spain coach praised his squad for remaining calm despite what he described as aggression and provocation from their opponents.
He said his players handled the situation with professionalism and prevented the clashes from becoming even more serious.
De la Fuente added that he was proud of the way his team responded during a difficult moment after securing Spain's World Cup triumph.
The confrontations were captured on video and quickly spread across social media, although no disciplinary action has yet been taken against any Argentina players or staff.
FIFA has confirmed it has opened an investigation into the incidents to establish whether any breaches of its disciplinary code took place during the post-match clashes.
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