Viewers of the video - in which the winger appears to insult Moroccans - reacted angrily to Elia's comments, forcing him to apologise and prompting Dutch coach Bert van Marwijk to impose the ban.
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Defender Gregory van der Wiel said on his Twitter account the squad were no longer allowed to use it, and this was confirmed by Ryan Babel and Elia who both went offline.
On the video Elia and Babel are seen playing a computer game in their room in front of a webcam, with several other team members visiting them.
Elia apologised for his comments but insisted he had not intended to insult a group of people.
"I want to apologise to the Moroccan community but I am not a racist," he told Dutch media.
"I grew up in The Hague in a neighbourhood with 75 percent Moroccans and have a lot of friends among them.
"What I said was meant for a friend of mine, Reduan, who always calls me 'negro'... it sounds odd, but it is some slang."
Van der Wiel last year learnt the hard way that Twitter comments can land you in hot water.
The defender pulled out of the Dutch squad for a friendly against Australia with concussion but said on Twitter that he had visited a concert the night the team travelled to Australia.
Netherlands start their World Cup campaign on Monday against Denmark.
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