TEL AVIV—Prime Minister Benjamin Netanyahu’s political future entered a treacherous new period Wednesday after he narrowly failed to form a coalition government and Israel’s parliament voted to dissolve and call a new election.

The fresh vote will force Israel back into campaign mode just weeks after Mr. Netanyahu narrowly won a presidential mandate following a bruising contest with an upstart party. The scenario is unprecedented, with all previous Israeli parliamentary elections since the country’s founding in 1948 leading...