The Shuhada Al-Aqsa hospital in central Gaza was hit by shelling, leaving five people dead -- one patient and four relatives, the Gaza Health Ministry said. Hamas TV showed upper floors damaged.
The Israel Defense Forces had no immediate comment. Israel has previously said militants use homes, schools, hospitals and mosques to launch attacks.
Israel killed more than 10 Hamas terrorists who entered the country through tunnels "to attack two different kibbutzim," or communal areas, "where farmers are trying to conduct their daily lives," government spokesman Mark Regev told CNN.
The death toll among Palestinians has passed 500. The biggest assault so far took place Sunday in the town of Shaja'ia.
On the Israeli side, where the Iron Dome defense system helps protect people against missile attacks every day, the death toll stands at 20 -- 18 soldiers and two civilians. Thirteen of the soldiers were killed Sunday in a Hamas attack. Two were Americans: California native Max Steinberg and Sean Carmeli, from South Padre Island, Texas, the U.S. State Department said.
"We will see Hamas come out of this substantially weakened, their arsenal of dangerous weapons diminished," Regev vowed Monday in an interview with CNN's Wolf Blitzer. "They will understand they can't shoot at our people with impunity."
But Palestinian leaders paint a very different picture. "Israeli massacres in Gaza result in mass civilian killings," official news agency WAFA, run by the Palestinian government in the West Bank, reported Monday. "At least 515 Palestinians have been killed," the news agency said.
It's unclear how many of the dead in Gaza were militants. The United Nations has estimated that 70% were civilians. Israel has reported that dozens of terrorists were killed in Gaza.
Israel blames Hamas for civilian deaths in Gaza, noting that the group has encouraged people to stay in their homes despite repeated warnings from Israel in advance of airstrikes. But some Palestinians have said they feared that even if they left they could face the same violence anywhere in Gaza. More than 83,000 Palestinians have taken refuge in U.N. facilities.
"Nobody is safe and nobody can flee anywhere because everywhere is targeted," said Enas Sisisalem, a mother of two who lives in the al-Remal neighborhood of Gaza City. "When we hear the shelling my kids will cry."
In a meeting late Sunday, U.N. Security Council members expressed "serious concern about the growing number of casualties," according to the body's president, Ambassador Eugene-Richard Gasana of Rwanda.
The members urged "an immediate cessation of hostilities" based on the cease-fire that stopped the 2012 conflict between Israel and Hamas, he said.