Israel-Gaza-Lebanon updates: Iranian president responds to Israel's retaliatory attacks

U.S. Secretary of State Antony Blinken met with Israeli Prime Minister Benjamin Netanyahu for over 2 1/2 hours on Tuesday, as the U.S. makes a push for a cease-fire agreement to end Israel's conflicts with Hamas and Hezbollah.

Blinken and Netanyahu discussed the need to capitalize on the killing of Hamas leader Yahya Sinwar to reach an agreement that would secure the release of the remaining hostages being held in Gaza and putting in place a plan that provides lasting peace for Palestinians and Israelis.

Blinken also discussed the importance of increasing the flow of aid being allowed into Gaza. The U.S. warned Israel last week that assistance could be withheld if humanitarian aid doesn't reach civilians in Gaza.

Lebanon's Rafik Hariri University Hospital will continue providing care despite the building sustaining "severe material damage" in an Israeli strike on Monday, according to Dr. Jihad Saadeh, the head of the hospital.

"Because of the targeting that has happened, we have sustained severe material damage to the hospital, including the destruction of solar panels, extensive destruction of all glass facades and damage to the hospital walls. Severe damage because it seems the shrapnel was very large," Saadeh said.

"We will not stop. Several entities even called me yesterday asking if I would like to evacuate the hospital. I told them, not at all. We will not evacuate. There is no hospital left but us. After the sinful targeting of the suburb’s hospitals, there is no one left except for us. So we will continue our work, God willing," Saadeh said.

The drone that targeted Israeli Prime Minister Benjamin Netanyahu's home on Saturday caused some damage to the structure, according to a person familiar.

Netanyahu called the attack an "attempt to assassinate me and my wife," in a statement Saturday. They were not in the home at the time of the attack, according to the prime minister's office.

-ABC News' Jordana Miller

At least 63 people were killed and 234 were injured in Israeli strikes on Lebanon on Monday, according to the Lebanese Ministry of Public Health.

Since the start of Israel's increased strikes on Lebanon, at least 2,546 people were killed and 11,862 others were injured, the ministry said.

-ABC News' Will Gretsky