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Sunday night's assault took place as thousands of people enjoyed inauguration festivities, raising new worries about Egypt's commitment to fighting sexual violence.
Authorities arrested seven men aged between 15 and 49 for sexually harassing women on Tahrir Square after the posting of the video, which caused an uproar in local and international media.
It was not clear whether the men arrested took part in the assault shown on the video.
"The Egyptian embassy in Washington DC and a number of Egyptian authorities, at the direction of President Abdel Fattah al-Sisi, have requested the YouTube administration to remove the video of the sexual assault victim," Sisi's spokesman said.
"This came in response to her wish, which she expressed during the president's visit to her yesterday at the hospital to check on her condition," he added in an emailed statement late on Thursday.
YouTube was not immediately available for comment on the Egyptian request. The clip showing the assault was still available on the video-sharing website on Friday.
Egypt approved a new law this month which punishes sexual harassment with at least six months in jail or fines of at least 3,000 Egyptian pounds ($420). The United States has urged Egypt to make good on its promises to fight sexual violence.
Sexual assault was rife at demonstrations during and after the 2011 uprising that ousted veteran president Hosni Mubarak and has been common for a decade at large gatherings in Egypt.
Sisi, Egypt's former army chief who won a landslide poll victory last month after deposing elected Islamist president Mohamed Mursi last July, has frequently spoken highly of women and their importance to society.
A police officer who rescued the victim of sexual harassment should be honoured, Sisi said, in an apparent reference to the woman in the video.
But some liberals have been wary of Sisi, especially after remarks he made defending an army practice - later denied by an army court - of conducting "virginity tests" on female protesters who complained of abuse.
Sexual harassment, high rates of female genital mutilation and a surge in violence after the Arab Spring uprisings have made Egypt the worst country in the Arab world to be a woman, a Thomson Reuters Foundation survey showed late last year.