The move came amid tension after the shooting of Jewish activist Rabbi Yehuda Glick.
Israel’s PM Benjamin Netanyahu called for calm, saying Mahmoud Abbas was responsible for escalating tensions.
Rabbi Yehuda Glick, a campaigner for greater Jewish prayer rights at the Temple Mount/al-Haram al-Sharif, was wounded.
Israeli police later killed a Palestinian suspected of shooting him.
The man, named as 32-year-old Moataz Hejazi, was shot after opening fire when police surrounded his home.
Rabbi Yehuda Glick is a well-known US-born campaigner for the right of Jews to pray at the site, which they are currently prohibited from doing. The compound is known to Jews as the Temple Mount, and to Muslims as Haram al-Sharif.
It is the holiest site in Judaism, and also contains the al-Aqsa Mosque – the third holiest site in Islam.
Palestinians hold the Israeli government responsible for a “dangerous act”, Mahmoud Abbas was quoted as saying by Nabil Abu Rudeina, in remarks carried by AFP news agency.
“This dangerous Israeli escalation is a declaration of war on the Palestinian people and its sacred places and on the Arab and Islamic nation,” Nabil Abu Rudeina added.
“The state of Palestine will take all legal measures to hold Israel accountable and to stop these ongoing attacks.”
However, PM Benjamin Netanyahu called for calm and suggested Mahmoud Abbas was responsible for the increasing tension.
“We’re facing a wave of incitement by radical Islamic elements as well as by the Palestinian Authority chairman… who said that Jews must absolutely be prevented from going on to the Temple Mount,” he said, quoted by Haaretz newspaper.
Benjamin Netanyahu added that reinforcements for the security forces would be brought into Jerusalem to keep order.
The shooting of Rabbi Yehuda Glick is the latest in a series of incidents which have led to an escalation of tensions in Jerusalem.
Some districts of East Jerusalem have seen nightly clashes between Palestinians and Israeli security forces since the conflict in Gaza.
Last week a Jewish baby and Ecuadorian woman were killed when a Palestinian attacker drove his car into a group of pedestrians at a tram stop in Jerusalem.
Police said Rabbi Yehuda Glick’s suspected attacker, Moataz Hejazi, had served time in jail in Israel and was released in 2012, adding that he belonged to the Islamic Jihad militant group.
The police anti-terrorist unit along with the Israeli internal security service Shin Bet had received information that Yehuda Glick’s attacker was located in the Abu Tor neighborhood, Israeli officials said.
Police say they were fired at after surrounding the house and shot back, hitting the suspect.
Rabbi Yehuda Glick has had surgery for gunshot wounds to his chest and abdomen.
He had just attended a conference where delegates discussed Jewish claims to the compound, one of the most contentious areas of the Arab-Israeli conflict.
Israel argues that it protects freedom of worship at the site but Palestinians claim it is unilaterally taking steps to allow larger numbers of Jewish visitors.
The site is administered by an Islamic body called the Waqf, while Israeli police are in charge of security.
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