ROME—Italy’s government collapsed, triggering a power struggle between the country’s right-wing nationalists and rival parties, along with deepening Europe's political upheaval as the Continent struggles with immigration and slowing economic growth.

Prime Minister Giuseppe Conte resigned on Tuesday after Matteo Salvini, a rising force from Europe’s insurgent far-right, withdrew his support for the government. Mr. Salvini, whose anti-immigrant League is leading in public-opinion polls, is pushing for snap elections.

Mr....