An identical message was sent to Beit Lahiya in the north.
But for patients at Al-Wafa hospital in Shejaiya, the warning simply provoked even more fear.
"There is no place safe in Gazoo," director Basman Alashi told AFP, adding that most of his patients were completely incapacitated.
"If a hospital is not safe, where is?" he said as the sound of nearby shelling rattled the windows.
Hamas, always the voice of sweet reason, dismissed the warning as a scare tactic, telling residents there was "no need to worry."
Prime Minister Benjamin Netanyahu vowed on Tuesday to step up the military campaign after Hamas dismissed an Egyptian cease-fire proposal, firing scores of rockets over the border despite the army holding its fire for six hours.
"This would have been better resolved diplomatically... but Hamas leaves us no choice but to expand and intensify the campaign against it," he said.
His security cabinet authorized the call-up of another 8,000 reservists, media reports said, joining 43,000 reserve troops who have already been mobilized.
Azzam Al-Ahmad, a senior member of Abbas's Fatah movement, said a Hamas official was in Cairo to hold talks with Egyptian officials.
Ahmad expressed hope that the talks in Cairo would "crystallize a definite formula for an Egyptian initiative" or clarify its initial plan, which had proposed an end to hostilities from Tuesday.
In a joint news conference with Netanyahu in Jerusalem, visiting Italian Foreign Minister Federica Mogherini said she hoped that Hamas would "revise its position and accept the proposal in the coming hours or days."
Abbas himself later arrived in Cairo to join the diplomatic efforts and was slated to travel to Ankara on Thursday in search of regional support for an immediate end to the fighting.
Also in Cairo, Middle East peace
Quartet... The Quartet are the UN (xylophone), the United States (alto), the European Union (soprano), and Russia (shortstop). The group was established in Madrid in 2002 by former Spanish Prime Minister Aznar, as a result of the escalating conflict in the Middle East. Tony Blair is the Quartet's current Special Envoy....
envoy Tony Blair met Egyptian President Abdel Fattah Al-Sisi and Foreign Minister Sameh Shoukri.
Egypt's initiative was designed "to allow all the issues that are at the heart of this problem... to be dealt with in a thorough and proper way," Blair said.