The meeting began with a moment of silence for dozens of Palestinians killed by Israeli forces on Monday, the bloodiest day there since a 2014 Israeli war on the Palestinian enclave.
Joanna Wronecka, Poland's ambassador, called for the gesture of remembrance in her role as current council president.
Kuwait had called for the session after Monday's killings.
The United States has already blocked the adoption of a UN Security Council statement that called for an “independent and transparent investigation” into Israel’s killing of Palestinian protesters on the Gaza border.
The statement, drafted by Kuwait ahead of a meeting on Tuesday, expressed “outrage and sorrow" at the deaths of at least 58 people during demonstrations over the opening of the American embassy in Jerusalem al-Quds.
It also demanded that all countries comply with a decades-old Security Council resolution calling on them not to station diplomatic missions in the occupied holy city.
Several Muslims and Western countries have denounced the violence against Palestinians.
Belgium on Tuesday demanded a UN inquiry into the violence in Gaza and summoned Israel's ambassador to the foreign ministry after she described all the Palestinian victims as "terrorists."
Belgium's foreign ministry said it summoned ambassador Simona Frankel over her remarks while Prime Minister Charles Michel said: "We call for an international inquiry led by the United Nations."
Germany says it supports calls for an independent investigation into the killing of dozens of Palestinians by Israeli forces during protests in Gaza.
Government spokesman Steffen Seibert told reporters in Berlin on Tuesday that the violence "concerns us greatly and it's terrible that so many people lost their lives, including minors."
Turkey has asked Israel's ambassador to temporarily leave Turkey.
Anadolu, Turkey’s official news agency news agency said Tuesday that the Turkish foreign ministry notified Eitan Na'eh, the Israeli ambassador that "it would be appropriate for him to return to his country for some time."
The ministry summoned the ambassador to protest Israel's use of deadly force on Palestinians and the U.S. decision to relocate its embassy there to Jerusalem al-Quds.
Turkey has also called home its ambassadors to Washington and Tel Aviv for consultations.
Israeli gunfire killed 59 Palestinians and wounded over 2,700 in the Monday clashes, the highest toll in a single day since a series of protests demanding the right to return to ancestral homes began on March 30.
The embassy inauguration also coincides with the climax of a six-week demonstration on the 70th anniversary of Nakba Day (Day of Catastrophe), May 15, when Israel was created.
The occupied territories have witnessed new tensions ever since US President Donald Trump on December 6, 2017 announced US recognition of Jerusalem al-Quds as Israel’s “capital” and said Washington would move US embassy to the city.
The dramatic decision triggered demonstrations in the occupied Palestinian territories as well as Iran, Turkey, Egypt, Jordan, Tunisia, Algeria, Iraq, Morocco and other Muslim countries.
The US ambassador to the United Nations on Tuesday told the emergency meeting on the violence in Gaza, that ally Israel had acted with restraint in the face of provocation from Hamas.
"No country in this chamber would act with more restraint than Israel has," Nikki Haley told the Security Council. "In fact the records of several countries here today suggest they would be much less restrained," she said.