TEL AVIV—Israel braced for fallout from a potential U.S. attack in Syria, as civilians flooded gas mask distribution centers and Prime Minister Benjamin Netanyahu held special consultations with defense chiefs and cabinet ministers on the Syria war at its border.

As Western governments step up accusations against Syrian President Bashar al-Assad of using chemical weapons against his opponents last week in a Damascus suburb, Israelis grow increasingly anxious that the Syrian leader could target the Jewish state in an act...