Tavecchio was confirmed in the new role on Monday, having won 63.33 per cent of a vote, beating off competition from former Italy and Milan midfielder Demetrio Albertini.
The election race was overshadowed by Tavecchio allegedly making a racial slur in the build-up to the ballot.
The 71-year-old reportedly made reference to "banana eaters" in response to the influx of foreign players in Serie A at a meeting of Italy's amateur leagues last month.
And former Milan, Juventus and Inter man Vieira - now in charge of Manchester City's elite development squad - has expressed his outrage at the appointment.
"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," he wrote on Twitter.
"For me, that shows how far away the Italian football authorities are from dealing with discrimination. 63 per cent of those that voted are admitting that they aren't fighting against racism, or that they don't want to fight these problems.
"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."