A family of six, including a five-year-old girl and a boy of three, were among 30 people killed in Israeli air strikes on Thursday.
Kerry has put the onus on Hamas to accept an Egyptian-proposed ceasefire deal.
He spoke by phone to the foreign ministers of Qatar and Turkey, hoping that both countries would use their influence to encourage Hamas to accept a ceasefire plan.
Khaled Meshaal, the Hamas political chief, is based in Qatar.
On Wednesday, Meshaal reiterated that his Palestinian group would reject any truce without the lifting of Israel’s eight-year blockade on Gaza.
“We will not accept any initiative that does not lift the blockade on our people and that does not respect their sacrifices,” he said.
Death toll rises
A senior Hamas official acknowledged to the Associated Press news agency that it was unrealistic to expect the blockade to end in tandem with a ceasefire and instead called for a firm agreement on principles on how to lift the siege.
Israel, which initially accepted a truce, has said it will keep up its military campaign as it eliminates tunnels that infiltrate the country from Gaza.
At least 730 Palestinians have been killed in the 17-day Israeli assault on the Gaza Strip. More than 4,600 people have been injured.
Two Israeli civilians have been killed by fire from Gaza since the offensive began.
The total number of Israeli soldiers killed since the start of the military assault rose to 32. One more soldier has been listed as missing and is believed to be dead.
Britain said on Thursday it was “gravely concerned” by the high number of civilian casualties resulting from Israel’s military operation in Gaza.
Foreign Secretary Philip Hammond said Britain would do everything it could to help broker a quick end to the hostilities.
Possible war crimes
At a news conference with Benjamin Netanyahu, the Israeli prime minister, Hammond expressed Britain’s support for Israel’s right to self-defence, adding that the current fighting was caused by Hamas firing rockets “indiscriminately” at Israeli towns and cities.
“But we are gravely concerned by the ongoing heavy level of civilian causalities,” he said.
The Gaza-based Palestinian Centre for Human Rights published figures on Wednesday showing more than 80 percent of the casualties were civilians, and a quarter of them children.
Netanyahu said Israel was doing everything it could to minimise casualties, pinning the blame on Hamas for using civilians as “human shields”.
But Valerie Amos, the UN humanitarian chief, said, in comments to BBC radio on Thursday, said: “It doesn’t matter how hard Israel tries to minimise harm, [Gaza] is an extremely overcrowded stretch of land.”
Her comments came a day after the UN Human Rights Council voted to launch an inquiry into the ongoing Israeli offensive’s alleged violations of international laws.
Navi Pillay, UN human rights chief, said there was “a strong possibility” that it was committing war crimes in Gaza.