Rice made the appeal at a news conference in the Saudi Arabian city of Jedda, before she left for Cairo on Tuesday to continue her tour of the Middle East.
“Innocent Palestinians are caught in this violence. I call on all parties to stop this violence,” Rice said.
However, there appears to be no end to the tension in the Palestinian territories, with the armed wing of Fatah threatening to kill senior members of rival group Hamas.
The al-Aqsa Martyrs’ Brigades made death threats to Khalid Mishaal, Hamas’s political chief; Said Siyam, the interior minister; and Yussuf al-Zahar, chief of Hamas’s paramilitary force in Gaza.
“We will charge ourselves with executing this sentence so that these scoundrels serve as an example to all those who dare spill Palestinian blood,” said the group in a statement.
For his part, Mishaal called three Arab leaders on Monday to talk to them about efforts to contain the Palestinian infighting.
He spoke to leaders of Qatar, Yemen and Sudan and urged Arab countries to live up to their responsibilities in lifting the siege on the Palestinians, the official Hamas website said.
Asked what to do about violent clashes between the Palestinian parties, Rice said: “The answer is for the Palestinians to find a government that can be committed to the Quartet principles.”
The Quartet grouping of the US, European Union, Russia and the UN backs the Middle East road map for peace, which calls on the Palestinians to renounce violence, recognise Israel and agree to abide by past peace agreements.
“The answer is for the Palestinians to find a government that can be committed to the Quartet principles”
The road map also calls for a Palestinian state to exist in peace and security alongside Israel, which must in turn end its illegal settlement programme in the occupied West Bank.
Saud al-Faisal, the Saudi foreign minister, speaking at a joint news conference with Rice, said the US administration should help restart peace talks to end the Israeli-Palestinian conflict.
Al-Faisal called the Israeli-Palestinian conflict a “core problem” and a “disease” that was encouraging violence in the region.
He expressed hope that Washington, a major ally of Israel, would help restart talks.
“It is the hope that … the United States … will restart the peace process and lead the region to peace and stability,” he said.
Rice then flew to Cairo for talks with foreign ministers from Egypt, Jordan and the six monarchies that make up the Gulf Cooperation Council (GCC) – Saudi Arabia, Bahrain, United Arab Emirates, Oman, Qatar and Kuwait.
Abdel Ilah Khatib, the Jordanian foreign minister, emphasised the importance of addressing the Palestinian issue before he flew to Cairo.
“Jordan and other Arab countries will insist that priority be given to solving the Palestinian question, which is at the core of the Middle East conflict,” Khatib said.
“The absence of a solution to this question is the cause of tensions and frustrations in the Middle East.”
Rice is also seeking to reinforce Arab support for Washington to place Iran before the UN Security Council over its attempt to obtain nuclear power.
Washington says Iran’s project concerns the manufacture of a nuclear bomb.
Tehran insists its use of nuclear technology is for civilian purposes.