Bush said he wanted to work with Abbas and Israel "to get a state defined".
"And I do so for a couple of reasons. One, it breaks my heart to see the vast potential of the Palestinian people really wasted," he said.
He said the second reason "is because it is the only way for lasting peace".
Bush also met Hosni Mubarak, Egypt's president, King Abdullah II of Jordan and Iraqi and Pakistani leaders.
King Abdullah "underlined the need for the United States to exert more efforts to prompt Israel to reach results in the negotiations with the Palestinians conducive to the establishment of an independent Palestinian state", a Saudi official said.
Bush's regional tour comes months after the US hosted the Annapolis conference aimed at restarting the stalled Middle East peace process.
He began his tour in Israel where he addressed the Knesset [parliament]as the Jewish state marked its 60th anniversary, prompting criticism that he is not acting like an honest broker in the peace effort.
"Bush has forgotten his role as the just mediator [of the Arab-Israeli conflict] and exposed his real self in the Knesset," an editorial in the Egyptian state-owned Al-Gomhuria newspaper said.
Bush called the close relationship between the US and Israel "unbreakable", and said that Israel is facing threats by adversaries in the Middle East.
Bush met Hamid Karzai, Afghanistan's president, on Saturday and will hold talks with Barham Ahmed Saleh, Iraq's deputy prime minister, on Sunday.