The woman walked up to the Henri Delaunay trophy, on display in the center of the Ukrainian capital Kiev, and knocked it over Saturday, ITAR-Tass reported.
She then displayed a crude message against the tournament. She was arrested by police.
The Femen group regularly stages topless protests and said Saturday's event was to highlight how Euro 2012 could increase prostitution in Ukraine.
"Femen is opposed to the hosting of Euro 2012 in Ukraine and accuses UEFA [Union of European Football Associations] of propaganda for sex tourism, of stimulating sex-tourism in the cities where the Euro tournament will be held, of debilitating the population and making it drunk," the group said in a statement on Twitter.
Poland and Ukraine are co-hosts for the Euro 2012 tournament, which runs from June 8 through July 1.