Palestinian officials said the blast at Al-Ahli al-Arabi hospital was caused by an Israeli air strike.
Israel blamed the blast on a failed rocket launch by the Palestinian Islamic Jihad group, which denied responsibility.
Following the NSC's appeal, Israeli airlines arranged flights from Istanbul late on Wednesday for Israelis who want to leave Turkey.
Protesters held fresh demonstrations near consulates of Israel and the United States in Istanbul on Wednesday evening.
Political analysts said the Gaza hospital blast could have dire consequences for ties between Israel and Turkey.
Persons:
Dilara, Israel, Tayyip Erdogan, Erdogan, Mert Ozkan, Huseyin Hayatsever, Daren Butler, Ali Kucukgocmen, Umit Bektas, Mehmet Emin Caliskan, Steven Scheer, Gareth Jones, Sandra Maler
Organizations:
REUTERS, Rights, Islamic, National Security Council, Istanbul, Police, Protesters, U.S, Embassy, NATO, Ece Toksabay, Bulent Usta, Thomson
Locations:
Al, Ahli, Gaza, Istanbul, Turkey, Rights ANKARA, Israel, Arabi, Ankara, Police, Turkey's, United States, U.S, Adana, Teneo, Jerusalem