"Promises from Hamas will not suffice," he said. "There must be monitoring mechanisms in place to help ensure that smuggling of weapons to terrorist groups in Gaza comes to an end."
Bush said Hamas was responsible for the latest violence and rejected a unilateral ceasefire that would allow Hamas to continue to fire mortars on Israel from the Gaza Strip.
But Riyad Mansour, the Palestinian Permanent Observer to the United Nations, told Al Jazeera that neither Hamas nor Israel was wholly to blame for the crisis.
"It's too simplistic to try to blame one side," he said, but added that the international community needed to act to halt Israel's bombardment.
"The first thing we need to do is to try and get a [Security Council] resolution that will halt this Israeli aggression."
Mahmoud Abbas, the Palestinian president, and several Arab foreign ministers are flying to New York over the weekend to urge the UN Security Council to adopt an Arab draft resolution that would condemn Israel and demand a halt to its bombing campaign in Gaza.
The US has said the draft is "unacceptable" and "unbalanced", because it makes no mention of halting the Hamas rocketing of southern Israel which Israel claims led to its offensive.
The US president's comments come after Condoleezza Rice, the US secretary of state, said on Friday that the US was working towards a "durable and sustainable ceasefire".
"We are talking constantly with the Israeli government to find a solution to Gaza that will be a sustainable one for the people of Gaza, for the people of Israel, and for the people of the Palestinian territories of the Middle East more broadly," she said in Washington DC.
"The first thing we need to do is to try and get a [Security Council] resolution that will halt this Israeli aggression"
Palestinian Permanent Observer to the UN
Bush said Israel's operations in Gaza were "instigated by Hamas - a Palestinian terrorist group supported by Iran and Syria that calls for Israel's destruction."
"The United States is leading diplomatic efforts to achieve a meaningful ceasefire that is fully respected. Another one-way ceasefire that leads to rocket attacks on Israel is not acceptable."
The US president called the heightened violence "an act of terror that is opposed by the legitimate leader of the Palestinian people, President (Mahmoud) Abbas."
More than 360 rockets have been fired from Gaza into Israel over the past seven days, killing four people.
Robert Serry, the UN's Middle East envoy, who said that Israel should open all crossings into the territory, but that this would "require a commitment from Hamas that all rocket attacks and smuggling will end".
"A movement like Hamas if it ... wants to serve the interest of its people, it will have to listen to the calls for an end to the violence and they will have to play a very important role in that,' he said.
Ban Ki-moon, the UN secretary-general, is to host talks on Gaza with several Arab leaders next week at the organisation's headquarters in New York next week, including Mahmoud Abbas, the Palestinian president.