Bush gave his first reaction to the stunning Hamas win in a White House news conference on Thursday.
He said the United States would not deal with Hamas, which comfortably won a Palestinian election victory, unless it renounced its call to destroy Israel.
"The United States does not support a political party that wants to destroy our ally Israel," Bush said.
"People must renounce that part of their platform.
"A political party that articulates the destruction of Israel as part of its platform is a party with which we will not deal.
"If your platform is the destruction of Israel, it means you're not a partner in peace. We're interested in peace," he said, addressing fears that the Hamas win would further stall US peace efforts in the Middle East.
Reflection of discontent
Bush portrayed the strong Hamas showing as more a reflection at discontent among Palestinians about how they were being governed than anger at the situation with Israel.
"It's a wake-up call to the leadership. Obviously, people were not happy with the status quo. The people are demanding honest government," he said.
Bush also said he would like US-backed moderate Palestinian leader Mahmoud Abbas to remain in office.
"We'd like him to stay in power," Bush said, when asked whether Abbas should remain in office in light of the vote results.
US Secretary of State Condoleezza Rice has said there could be no Middle East peace process if Hamas, winner of the Palestinian election, refused to recognise Israel's right to exist.
Rice said peace process would be
hit if Hamas stuck to its position
"You can't have a peace process if you're not committed to the right of your partner to exist," she told the World Economic Forum in Switzerland on Thursday, speaking by videolink.
"And I think you will hear the international community speak clearly on exactly those principles over the next day. There will be some difficult choices before those in whom the Palestinian people are placing their trust."
She was speaking after unofficial election results on Thursday showed a shock victory for the Islamist group Hamas over the long-dominant Fatah faction. Official results were due around 1700 GMT.
Rice said the election had been peaceful and fair, with very high turnout. But she said the US position on Hamas, which it considers a terrorist organisation, was unchanged.
"As we have said, you cannot have one foot in politics and the other in terror. Our position on Hamas has therefore not changed."
Rice added: "Anyone who wants to govern the Palestinian people and do so with the support of the international community has got to be committed to a two-state solution, must be committed to the right of Israel to exist.
"As we have said, you cannot have one foot in politics and the other in terror"
US Secretary of State
"But if there is to be a future that can answer the aspirations for peace of the Palestinian people ... then it is going to have to be a future that renounces violence and terrorism."
Rice also voiced support for Palestinian President Mahmoud Abbas.
"She told President Abbas the US administration will continue to back him and his policies," presidential spokesman Nabil Abu Rdainah said.
"She listened to his point of view and they agreed to continue contacts."