Israeli Prime Minister Benjamin Netanyahu warned the army would "expand and intensify" its Gaza operation after Hamas rejected the Egyptian truce proposal.
Read: Key demands, issues in Israel-Gaza ceasefire
The renewed Israeli strikes killed two Gazans, raising the Palestinian death toll in eight days of violence to 196, medics said.
The Israeli was killed in a rocket attack on an Israeli position near the Erez crossing with Gaza, the army said.
Hamas's Ezzedine al-Qassam Brigades armed wing said it carried out the attack.
The 38-year-old civilian had been delivering food to soldiers serving in the area, a spokesman for the Israeli emergency services told AFP.
Read: Gaza death toll hits 196 on eighth day of violence
A picture from the southern Israeli Gaza border shows a rocket being launched from the Gaza strip into Israel. (AFP photo)
It was the first Israeli death of the conflict after nearly 1,000 rockets and mortar rounds hit the Jewish state. Four Israelis have been seriously wounded.
It came after Israel's security cabinet said early Tuesday it would accept an Egyptian proposal for a ceasefire to begin at 0600 GMT.
But Hamas officials said they had not been consulted on the proposal and would not halt fire without a full-fledged deal including Israeli concessions.
The movement's armed wing continued to fire dozens of rockets into Israel after the 0600 GMT deadline, sending tens of thousands scrambling for cover.
At 1200 GMT, the Israeli army announced it was resuming air strikes, after militants fired 47 rockets from Gaza.
The fresh raids hit Gaza City, southern Khan Yunis and Rafah and killed two people.
"This would have been better resolved diplomatically, that's what we tried to do when we accepted the Egyptian truce proposal today," Netanyahu said.
"But Hamas leaves us no choice but to expand and intensify the campaign against it," he added.
Overnight, Hamas's Ezzedine al-Qassam Brigades armed wing rejected the Egyptian proposal for a truce to be followed by talks.
Hamas spokesman Sami Abu Zuhri said the movement had not been consulted on the truce bid, and called the idea of halting fire before agreeing on terms "unacceptable".
A top member of Hamas's exiled politburo, Mussa Abu Marzuq, sounded a more cautious note, saying the movement had no official position on the proposal and discussions were continuing.
Palestinian relatives mourn over the body of Noor Alnajedy (10) who was killed during an Israeli air strike on a building near his family house, at the main mosque during his funeral in Rafah refugee camp, southern Gaza Strip. (AP photo)
Hamas has said it wants the end of Israel's blockade of Gaza and the opening of the Rafah border crossing with Egypt as part of a truce deal.
It also wants Israel to free Palestinians it rearrested after releasing them in a 2011 exchange for an Israeli soldier held by Gaza militants for more than five years.
'Cool and with patience'
In his remarks on Tuesday evening, Netanyahu also took aim at domestic critics, responding to Foreign Minister Avidgor Lieberman who earlier slammed Israel's acceptance of the Egyptian truce and Netanyahu's "hesitance", saying Israel should conquer Gaza and expel Hamas from it.
"These are moments when decisions must be made coolly and with patience, not hastily or noisily," Netanyahu said.
The Israeli premier also fired deputy defence minister Danny Danon, a firebrand member of his Likud party, who was a vocal critic of him during the operation.
Cairo's truce proposal was announced overnight, and urged both sides to halt the violence and travel to Egypt for talks.
Palestinian president Mahmud Abbas was due in Cairo on Wednesday, but it was unclear if Hamas officials there were continuing to discuss the truce bid and if Israeli officials would also travel to Egypt.
The proposal won support from Western governments with President Barack Obama saying he was "encouraged" by Egypt's efforts and hoped to see calm restored.
German Foreign Minister Frank-Walter Steinmeier also urged Hamas to accept the Egyptian proposal, accusing the Islamists of holding Gaza "hostage."
But Turkish Prime Minister Recep Tayyip Erdogan put the blame entirely on Israel accusing it of carrying out "state terrorism" and a "massacre" of Palestinians in Gaza.
Smoke billows from buildings following an Israeli air strike in Gaza City. (AFP photo)
Strikes resume across Gaza
Israel launched Operation Protective Edge before dawn on July 8, hitting Gaza with an intensive air and artillery bombardment aimed at stamping out rocket fire.
Since then, 960 rockets have hit Israel, while another 215 have been intercepted by its Iron Dome air defence system, the army said.
In Gaza City on Tuesday, shortly after Israel resumed its air strikes, 44-year-old Suheil al-Hossari looked at the ruins of his home.
An Israeli warning call meant there were no deaths in the strike, but that was small comfort for Hossari.
"Everything is destroyed. The food we prepared for our iftar (Ramadan evening meal) is now under the ruins of my home," he told AFP.
"I don't have money for food, let alone to rent a new house. I will stay here in the open and rely on God's mercy."
Read: Obama praises Egypt's ceasefire plan for Israel
(With inputs from Reuters and AP)