Seventy-one-year-old Tavecchio was put in office on Monday, winning 63.63 per cent of votes cast in the hunt for Giancarlo Abete's replacement.
His name has been marred by allegations of racism, though, relating to supposed comments made about African players eating bananas.
Tavecchio has denied any wrongdoing but Vieira - born in Senegal and who played for three Italian clubs during an illustrious career - thinks his election sends out the wrong message.
Writing on Twitter he said: "I'm finding it really hard to believe that Carlo Tavecchio has been elected as president of the Italian FA after the comments that he made. For me, that shows how far away the Italian football authorities are from dealing with discrimination."
The Manchester City director of elite development added: "I played in Italy for years so I know the issues, I saw the issues.
"If he was an English man making this kind of comment, politically, he would be completely out.
"This decision was made in football, but I think it is bigger than football. It should make the whole country of Italy look at the message they want to send about what they think about racism.
"I can't believe he will represent the Italian Football Authorities. What a shame."
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