Israel says it “regrets” the deaths of five Egyptian policemen on the border with Gaza.
The country’s Defence Minister Ehud Barak said he had told the army to hold a joint investigation with Egypt into Thursday’s clashes.
It is thought the policemen were shot by Israeli forces as they pursued cross-border raiders.
Egypt has recalled its ambassador from Tel Aviv for the first time in ten years, accusing Israel of breaching the peace treaty between the two countries.
Several hundred Egyptians kept up a protest at the Israeli embassy in Cairo, demanding the expulsion of the Israeli envoy.
One man said: “If Israel thinks that things are like the days of the corrupt regime, when Egyptian soldiers were killed and we were forced to remain quiet, they are wrong. There has been a revolution and I want to tell Israel that one Egyptian soldier is worth 100 Israeli soldiers.”
Cairo is also demanding an apology from Israeli leaders who have accused Egypt of neglecting its border.
One analyst says Egypt’s new assertiveness has put Israel on the back foot.
“Egypt will not get back to where it was in the Mubarak era so I think what Israel should do and should learn from this incident is that this is going to become kind of a different Middle East,” said Professor Uzi Rabi, Director of the Dayan Centre for Middle Eastern studies at Tel Aviv University.
Renewed Israeli airstrikes on the Gaza Strip have prompted the Arab League to hold an emergency meeting on Sunday.
Israel’s retaliation follows Thursday’s attack blamed on Palestinians near the Red Sea resort of Eilat, which killed eight Israelis.
One strike was aimed at a militant riding a motorbike in Gaza. Hamas says the missile also struck a car, killing three civilians including a doctor and a child.