In phone calls with Abbas and Jordan's King Abdallah, Macron expressed France's "serious concern about the situation in Gaza, in Jerusalem and the Pälestinian towns", a statement by the French presidential palace said.
"He deplored the large number of Palestinian civilian victims in Gaza today [Monday] and in the last few weeks," the statement went on. "He condemned the violence of the Israeli armed forces against the demonstrations."
At least 58 Palestinians, including six minors, were killed and 2,700 wounded, 1,360 by bullets, according to Gaza local authorities, as Israel marked the 70th anniversary of its creation and celebrated the transfer of the US embassy to Jerusalem.
Macron expressed "disapproval" of the move, saying that the status of Jerusalem, a divided city that both Israelis and Palestinians claim as their capital, had to be decided "between the parties in a negotiated framework under the aegis of the international community".
He expressed support for the "Palestinians' right to peace and security", as well as Israel's security and "France's consistent position in favour of two sates, Israel and Palestine, living side by side with secure and recognised borders".
US blocks UN resolution
The United States yesterday blocked the adoption of a UN Security Council statement that would have called for an independent probe into the violence.
Abbas - alongside Egypt and Syria - accused Israel of "massacres".
Abbas said the new embassy was tantamount to "a new American settler outpost" in Jerusalem and that the United States "is no longer a mediator in the Middle East".
Kuwait requested an emergency meeting of the UN Security Council on Tuesday and condemned the bloodshed.
The surge in fatalities led South Africa to recall its ambassador in Israel "with immediate effect until further notice".
Russia, Belgium and Morocco all denounced Washington's "unilateral decision" to move its embassy.
EU foreign policy chief Federica Mogherini called for all sides to act with utmost restraint to avoid further loss of life, while British Prime Minister Theresa May's spokesman urged "calm and restraint".
Turkish President Recep Tayyip Erdogan meanwhile accused Israel of "state terror" and "genocide", announcing three days of mourning and a giant protest in Istanbul on Friday.
Turkey also said it was recalling its ambassadors to the US and Israel "for consultations".
Amnesty International called the violence an "abhorrent violation" of human rights.
Human Rights Watch issued a statement condemning a "bloodbath".
The six weeks of protests in Gaza culminate on Tuesday with the commemoration of what Palestianians call the "Nakba" (catastrophe), which saw 700,000 Palestininans flee or be expelled from their homes in the 1948 war surrounding Israel's creation.