Amid rocket attacks by the militant group Hezbollah into Israel and Israeli bombardment around Beirut, Secretary of State Antony J. Blinken began a tour of the Middle East on Tuesday, making renewed calls for a cease-fire in the Gaza Strip and a diplomatic solution to the escalating conflict in Lebanon.

Meeting with Prime Minister Benjamin Netanyahu in Jerusalem, Mr. Blinken pressed Israel “to capitalize on” the killing last week of Hamas’s leader, Yahya Sinwar, and to end the war with Hamas in Gaza, a State Department spokesman, Matthew Miller, said in a statement.

On his 11th trip to the Middle East since the conflict began a little more than a year ago, Mr. Blinken met with Mr. Netanyahu for two and a half hours. His visit was bracketed by the sounds of air-raid sirens in multiple locations across Israel during the morning and afternoon as the military tracked what it called Hezbollah “projectiles” coming from Lebanon. The morning attack, on an Israeli military base near Tel Aviv, sent residents fleeing into shelters but caused no casualties or significant damage, officials said.

In Beirut, the Lebanese capital, rescue workers pulled bodies from the rubble of buildings hit overnight by the Israeli military, killing at least 18 people, including four children, according to Lebanon’s health ministry. The buildings were near Rafik Hariri University Hospital, the largest public health center in Lebanon, which was also damaged.

On Tuesday afternoon, a separate Israeli airstrike leveled a building in the Dahiya area, south of Beirut, minutes after a Hezbollah spokesman briefed reporters nearby. The Israeli military described the target as Hezbollah intelligence headquarters, with 25 Hezbollah members present.