Smoke rises above building following an Israeli air strike on Gaza City, on August 6, 2022. AFP
National Security Council spokesman John Kirby urged calm from both sides but said US ally Israel had the right to defend itself.
"We certainly urge all sides to avoid further escalation... We absolutely fully support Israel's right to defend itself against terrorist groups that are taking the lives of innocent civilians in Israel," he said.
Russia, whose ties with Israel have been tested by Moscow's invasion of Ukraine, appeared to pin initial blame on Israel.
"We are observing with profound worry how events are evolving," foreign ministry spokeswoman Maria Zakharova said, adding that Moscow was calling "on all the parties involved to show maximum restraint".
"The new escalation was caused by Israeli army firing into the Gaza Strip on August 5, to which Palestinian groups responded by carrying out massive and indiscriminate bombardments on Israeli territory," she said.
Britain called for "a swift end to the violence", while reasserting its support of Israel.
"The UK stands by Israel and its right to defend itself," Foreign Secretary Liz Truss tweeted.
"We condemn terrorist groups firing at civilians and violence which has resulted in casualties on both sides."
Health authorities in the Palestinian enclave have said a five-year-old girl was among 15 people killed in Israeli bombardment.
No deaths have been reported on the Israeli side.
The EU called for maximum restraint, as Israel warned Saturday that its air strikes could last a week and as cross-border fire reverberated for a second day, in the worst escalation since the last Gaza war in 2021.
"The European Union follows with great concern the latest developments in and around Gaza," a spokesman for the EU's foreign policy chief Josep Borrell said.
"The EU calls for maximum restraint on all sides... While Israel has the right to protect its civilian population, everything must be done to prevent a broader conflict," he said.
UN Middle East peace envoy Tor Wennesland said he was "deeply concerned", warning that the escalation was "very dangerous".
Egypt has reportedly been seeking to mediate and may host a delegation from Islamic Jihad over the weekend.
The group has been firing rockets from Gaza after one of its commanders was killed in an Israeli strike on Friday.
The Arab League "condemned in the strongest possible terms the ferocious Israeli aggression against Gaza".
Jordan's foreign ministry has "stressed the importance of halting the Israeli aggression".
Iran, a supporter of Gaza militants, said the territory was "not alone" in its fight.
"We are with you on this path until the end, and let Palestine and the Palestinians know that they are not alone," Iran's Revolutionary Guards chief Major General Hossein Salami told Islamic Jihad leader Ziad al-Nakhala during a meeting in Tehran.
Football friendly called off
The violence has also affected the sporting scene in Israel, where sirens blared to warn of incoming rocket fire from the Gaza Strip.
A pre-season friendly between Spanish side Atletico Madrid and Italian giants Juventus that had been scheduled to take place in Tel Aviv on Sunday was cancelled because of "security" concerns, the two teams announced.