"The secretary reiterated his call on all parties to de-escalate tensions and bring a halt to the violence", said a State Department statement.
In the Israeli city of Lod, a 52-year-old Arab Israeli and his 16-year-old daughter were killed early Wednesday when a rocket landed in the courtyard of their one-story home.
The Israeli raids on Gaza come after weeks of mounting tension over the looming forced expulsion of Palestinian families from Sheikh Jarrah, a Palestinian neighbourhood in occupied East Jerusalem that Jewish settlers have been trying to expel them from for decades.
Reinforcements were sent to the area after clashes broke out the night after an Israeli Arab was killed in the city.
Israel's Defence Minister Benny Gantz vowed more attacks on Palestinian militant groups in Gaza to bring "total, long-term quiet" before considering a ceasefire.
Since the beginning of the Muslim holy month of Ramadan in mid-April, Palestinians have faced off nightly with Israeli police in East Jerusalem, who put up barriers to stop evening gatherings at the walled Old City's Damascus Gate. A 26-year-old Palestinian was killed Tuesday in clashes with Israeli troops in a refugee camp near the city of Hebron.
"We believe Palestinians and Israelis deserve equal measures of freedom, security, dignity and prosperity, and USA officials have in recent weeks spoken candidly to Israeli officials about how evictions of Palestinian families who have lived for years, sometimes decades, in their homes and how demolitions of these homes work against our common interests and achieving a solution to the conflict", she said.
The United States sees the Security Council meeting as a sufficient show of concern, one diplomat said on condition of anonymity.
The U.N. Security Council is planning to hold an emergency closed-door meeting Wednesday to discuss the situation.
With the Israeli deaths, along with 10 injuries, Prime Minister Benjamin Netanyahu said officials have made a decision to "increase both the strength and rate of the strikes" on the Gaza Strip against the militant groups Hamas and Islamic Jihad.
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"Hamas will receive blows now that it didn't expect", he said.
The violence quickly spread to the Old City compound containing Al-Aqsa Mosque, the third holiest shrine in Islam and the most sensitive site in the Israeli-Palestinian conflict.
In a statement, the armed wing of Hamas said Issa was killed "along with a few of his fellow brothers of leaders and holy fighters" during the fighting that has been going on for two days in Gaza. Israeli Prime Minister Benjamin Netanyahu warned that militants would pay a "very heavy" price for the rockets, which reached the outskirts of Jerusalem on Monday during a holiday in Israel commemorating its capture of East Jerusalem in a 1967 war. We will not tolerate attacks on our territory, our capital, our citizens and our soldiers.
Hesitation from the USA mission prevented the release of the statement after the first meeting earlier this week, though at the time Washington said it wanted more time to deliberate.
Bassem Issa is the highest-ranking military figure in Hamas to be killed by Israel since 2014, according to The Associated Press.
The proposed statement called on Israel "to cease settlement activities, demolitions and evictions, including in east Jerusalem in line with its obligations under worldwide humanitarian law" and refrain from unilateral steps "that exacerbate tensions and undermine the viability of the two-state solution". However, the court postponed a key ruling in the case Monday, citing the "circumstances".
Violence has also broken out in mixed Arab-Jewish cities across Israel, with members of Israel's 21 per cent Arab minority angry over the Jerusalem evictions and Gaza violence.
Israel considers all of Jerusalem as its unified capital.
British Foreign Secretary Dominic Raab condemned the rocket attacks, saying, "The ongoing violence in Jerusalem and Gaza must stop".
The US mission to the United Nations was not immediately available for comment.