Incursion 'a massacre'
Israel's incursion came a week after Bush said he wanted a peace deal between the Israelis and the Palestinians by the end of his presidential term at the end of 2008.
Hamas officials blamed Bush's presence in the region for the violence.
Ehud Olmert, the Israeli prime minister, has reportedly ordered a series of such "sharp and short" incursions, into Gaza and the West Bank.
Abbas said on Tuesday: "What happened today is a massacre, a slaughter against the Palestinian people.
"Our people cannot keep silent over these massacres. These massacres cannot bring peace."
Peace bid 'damaged'
He said the Gaza operation, in which Hussam al-Zahar - the son of Mahmoud al-Zahar, a former Hamas foreign minister - was also killed had severely damaged the peace efforts relaunched by Bush in the US city of Annapolis less than two months ago.
It is the second of al-Zahar's sons to be killed by Israel.
Al Jazeera's Jacky Rowland, reporting from Gaza, said that the death toll was likely to rise.
She said: "It's not just the destruction and devastation, but the effect this operation will have on peace talks."
The US president had started his tour last week by visiting Israel and emphasised America's bond with Israel.
Ahmed Qureia, the chief Palestinian negotiator, said that if the violence continued it would render the negotiations "meaningless".
Taher Nunu, a Hamas spokesman, said: "What is clear is that there is an Israeli plot to destroy the whole region.
"President Bush gave the green light to such aggression and such terrorism."
He said: "To all the Arab countries, to the custodian of the two holy mosques [King Abdullah of Saudi Arabia], Arab leaders and heads of states, we would like to tell them that they have to act immediately in order to put an end to this aggression."
Israel said they were engaging in an operation "against terror threats".