Arrests. Classified documents. And suspected leaks that may have harmed efforts to free hostages held by Hamas in order, critics say, to give Benjamin Netanyahu public cover for failing to agree to a cease-fire deal. The Israeli prime minister was engulfed in scandal Monday over a case involving one of his aides that has sent shockwaves across the country.

The firestorm — brought into public view when an Israeli court loosened a gag order Sunday night — has enraged Netanyahu’s political…