JERUSALEM—Benjamin Netanyahu’s deadline for forming a new Israeli government expired early on Wednesday, with the country’s longest-serving prime minister having failed to break more than two years of political deadlock. There was also no guarantee that, after the conservative incumbent was unable to assemble a new coalition, parties outside his caretaker government could bridge their differences and unseat him. Netanyahu, 71, has been in office since 2009 and also served for three years in the 1990s. He has been fighting to hold the helm through four inconclusive elections since 2019. He is on trial for criminal corruption charges that he denies. With the midnight deadline having passed, President Reuven Rivlin can assign the coalition-building task to another member of parliament. That is widely expected to be Yair Lapid, 57, whose centrist Yesh Atid party placed second to Netanyahu’s Likud in the March 23 vote. Netanyahu’s bloc of right-wing and Jewish …