"We do have in the Palestinian territories a government that is devoted to the international principles, the foundational principles for peace and this is an opportunity that should not be missed," Rice said ahead of her visit.
When Hamas wrested control of Gaza from Abbas' Fatah forces in mid-June, the president of the Palestinian Authority fired the group from the government and installed a new cabinet in the West Bank.
"Ultimately, the Palestinian people will have to choose what kind of world they will live in, what kind of state they will have," Rice said, at the end of a four-day Middle East trip.
The tour is partly aimed at laying the groundwork for an international peace conference that the United States will host later in the year.
Rice is expected to discuss with Abbas the prospect of bolstering forces in the West Bank to counter Hamas and prepare for any Israeli withdrawal.
Rice and Abbas are also scheduled to sign a "framework agreement for security assistance" on Thursday.
The move follows Washington's recent pledge of at least $86m to Abbas, officials said.
Other Palestinian concerns, such as release of tax arrears to the Palestinian leadership and the easing of severe Israeli-imposed security restrictions, are also expected to be discussed, a US official said.
Abbas received a boost this week after Moscow downgraded its contacts with Hamas, while maintaining some ties with the group to foster dialogue.
The decision came after Abbas met with Vladimir Putin, the Russian president, in Moscow which, according to a Russian official, has taken a dim view of Hamas's use of force to take control of Gaza.
A US official cited "very interesting" reports about the Abbas-Putin talks in an indication that Rice would be anxious to get details of the Moscow encounter from the Palestinian leader.
Rice travelled with Robert Gates, the US defence secretary, to the Middle East.
They travelled together to Egypt and Saudi Arabia before Gates went on to Kuwait while Rice headed to Israel and the Palestinian territories.