The Israeli military said that since midnight it had attacked more than 30 sites in the coastal enclave where Hamas Islamists are dominant, without specifying the targets.
Gaza militants fired six rockets at towns in Israel's south on Saturday setting off alarm sirens and causing no damage or injuries, a military spokeswoman said.
Violence also picked up in the occupied West Bank, the Palestinian territory where President Mahmoud Abbas's Fatah movement holds sway, where a Palestinian man, 43, died of a gunshot wound to the chest from a confrontation with Israeli soldiers in the city of Hebron, medical officials said.
Israeli troops shot and killed another Palestinian man, 20, on Friday at a protest near a Jewish settlement outside Ramallah, Israeli military officials said.
Egypt, helped by American and European mediators, has made no visible progress toward resuming a 72-hour ceasefire that halted the fighting between Israel and Gaza militants that began on July 8, but was expected to pursue these efforts again on Saturday.
The ceasefire expired on Friday with the sides still far apart on the terms for renewing the deal and each side blaming the other for refusing to extend it.
Israel accused Hamas of firing several rockets about four hours before the deal expired at 8 a.m. (0500 GMT). In all, Gaza militant fired 57 rockets at Israel throughout Friday.
Israel also launched air strikes in Gaza on Friday, killing five Palestinians, among them a 10-year-old boy near a mosque in Gaza City. An Islamic Jihad militant and three other Palestinians were killed in the southern Gaza Strip.
Police said two people in Israel were injured by mortar fire from Gaza on Friday.
Ramping up the pressure
By resuming attacks against Israel, Gaza militants appeared to be trying to ramp up pressure and making it clear they were ready to fight on to fulfil a goal of ending a blockade of the territory that both Israel and neighboring Egypt have imposed.
Heavy civilian casualties and destruction during Israel's campaign against militants in packed residential areas of the Gaza Strip have raised international alarm over the past month, but efforts to prolong a ceasefire at talks in Cairo failed.
Gaza officials say the war has killed 1,886 Palestinians, most of them civilians.
Israel says 64 of its soldiers and three civilians have died in the fighting that began on July 8, after a surge in Palestinian rocket salvoes into Israel.
It expanded its air and naval bombardment of the Gaza Strip into a ground offensive on July 17, and pulled its infantry and armour out of the enclave on Tuesday after saying it had destroyed more than 30 infiltration tunnels dug by militants.
The White House urged Israel and the Palestinians to do what they could to preserve civilians after having failed to extend their ceasefire. Spokesman John Earnest said on Friday "the United States is very concerned" about the renewed violence.
"We condemn the renewed rocket fire and we are concerned about the safety and security of civilians on both sides of the conflict," Earnest said.
U.N. Secretary-General Ban Ki-moon issued a similar statement urging the parties "not to resort to further military action that can only exacerbate the already appalling humanitarian situation in Gaza."
Israel had earlier said it was ready to agree to an extension as Egyptian go-betweens pursued negotiations with Israeli and Palestinian delegates.
Hamas did not agree. Hamas spokesman Sami Abu Zuhri said Israel had rejected most of the group's demands. He said the Palestinians had wanted Israel to agree in principle to lift a Gaza blockade, release prisoners and permit the opening of a sea port, but these had been rebuffed.
"However, we did not close the door and will continue with the negotiations," Abu Zuhri said.
Israel has shown little interest in easing its naval blockade of Gaza and controls on overland traffic and airspace, suspecting Hamas could restock with weapons from abroad.
Israeli Justice Minister Tzipi Livni, a member of Prime Minister Benjamin Netanyahu's security cabinet said the issue of a sea port should be part of wider peace negotiations with the Palestinians and that Hamas should not at this time be rewarded for "using force against Israeli citizens."
In Cairo, the foreign ministry called for a resumption of the ceasefire, saying only a few points remained to be agreed.
Egypt mediates the talks but is meeting separately with each party. Israel and Hamas deny each other's legitimacy, with Hamas rejecting Israel's right to exist and Israel rejecting Hamas as a terrorist organisation.