NEW YORK—Italy won't successfully push back against austerity policies in Europe until it shows it is credible in driving through economic overhauls at home, Prime Minister Matteo Renzi said Wednesday.

The decisive showing by Mr. Renzi's center-left party in May's European elections left him one of the Continent's strongest leaders, propelling him to the forefront of a growing campaign to relax austerity measures to foster economic growth.

...