Bush told Hamas: "You must stop Gaza from being a safe haven for attacks against Israel. You must accept the legitimate Palestinian government," allow aid to Gaza, disarm militias, and recognise Israel.
Bush also urged Arab nations to end "the fiction that Israel does not exist", curb anti-Israel rhetoric in their media and send cabinet-level officials to the Jewish state.
And he pushed Olmert to continue to release Palestinian tax revenue to Salam Fayyad, the Palestinian prime minister, halt Israeli settlement expansion and dismantle unauthorised outposts.
Commitment to Israel
Israelis "should be confident that the United States will never abandon its commitment to the security of Israel as a Jewish state and homeland for Jewish people", he said.
|Bush telephoned Abbas to reaffirm |
US support for the president [AP]
Bush also hailed the talks between Olmert and Abbas, as a senior Israeli official said that the prime minister had pledged to free 250 Palestinian prisoners in a goodwill gesture to the Palestinian leader.
But while welcoming the release, the Palestinians said the freeing of 250 prisoners out of the more than 11,000 held in Israeli jails was not enough.
Bush also announced a direct US contribution of $80m to help Abbas reform his security services. Two US officials said the money was being shifted from Gaza to Fayyad's government.
More US aid will come when former British prime minister Tony Blair, now the envoy for the Middle East "Quartet" of the US, UN, European Union, and Russia, reports success in building a plan for bolstering Palestinian security and political institutions.
"With the proper foundation, we can soon begin serious negotiations toward the creation of a Palestinian state," said Bush.
The Quartet is to meet in Lisbon, the Portuguese capital, on Thursday to take stock after Abbas appointed a new government following the takeover of Gaza by Hamas.
Meanwhile Bush telephoned Abbas on Monday to reaffirm Washington's support for the Palestinian president, a spokesman for Abbas said.
The two men spoke after Bush called for an international conference to revive Middle East peace talks.
"President Bush expressed his support for president Abbas and for efforts to relaunch the peace process through this conference" in the autumn,
Nabil Abu Rudeina told reporters.
"President Bush also expressed his support for the Palestinian government led by Salam Fayyad," he added.
But some analysts say the strategy to back Abbas and isolate Hamas could backfire if Abbas is seen as a collaborator with Israel and Washington.