TEL AVIV — In everything but name, it’s a war.

As an increasingly nervous West looked on, Israeli forces and Hezbollah gunners in Lebanon traded fire Thursday in the fiercest regional flare-up in a decade.

Determined to retrieve two soldiers abducted by Hezbollah in a cross-border raid Wednesday, in which eight other soldiers died, Israel shelled targets as far north as Beirut, killing dozens of people. Israel also bombed Lebanon’s main airport and imposed a naval blockade on the coast.

Scores of Hezbollah rockets and mortar bombs rained down on northern Israel, hitting Carmiel, Safed and even Haifa, Israel’s third-largest city. One woman was killed and dozens were injured.

The woman was identified as Monica Saidman, 47, killed when a rocket hit her apartment in Nahariya. A spokesman for the Jewish Agency for Israel in Argentina told JTA that Saidman moved to Israel three years ago.

Another rocket landed in an immigrants absorption center in Safed, injuring an immigrant from Ethiopia, according to the Jewish Agency.

Tens of thousands of residents of northern Israel hid in shelters or fled.

“It’s not just the two soldiers who are hostages: We are all hostages,” a Carmiel resident shouted from his balcony as rescue units scoured an area struck by a Katyusha rocket.

Lebanon called for a ceasefire, but Israel demanded that Lebanon finally comply with U.N. Security Council resolutions ordering it to disarm Hezbollah.

“The Syrians and Iranians have allowed Hezbollah free rein in recent years. Lebanon today is paying the price of its dereliction, which enabled Hezbollah to behave as if Lebanon belongs to it,” Defense Minister Amir Peretz told the Knesset Foreign Affairs and Defense Committee. “Through this operation, we intend to prevent Hezbollah from returning to the border.”

While Israel stopped short of declaring war on Lebanon, officials said there would be no let-up in the offensive.

The northern offensive began Wednesday after three Israeli soldiers were killed and two were seized while on border patrol. Israeli forces were quick to respond, as artillery units and warplanes shelled bridges in southern Lebanon that could be used to smuggle the captive soldiers northward.

On the ground, armored columns rolled across the border and engaged in running battles with Hezbollah gunmen. At least two Lebanese were killed. An Israeli tank hit a mine, killing four other Israeli soldiers; the eighth was killed while trying to rescue his comrades from the burning tank.

The kidnapped soldiers were identified as Ehud Goldwasser, 31, of Nahariya, and Eldad Regev, 26, of Kiryat Motzkin. Their medical conditions were unknown.

Prime Minister Ehud Olmert, already involved in a military sweep of the Gaza Strip aimed at retrieving a soldier abducted there by Palestinian gunmen last month, received a boost from President Bush.

“Israel has the right to defend herself,” Bush told reporters during a visit to Germany.

The European Union took a different tack, calling Israel’s retaliation against Lebanon “disproportionate.”

While assailing Syria, one of Hezbollah’s main patrons, Bush urged Israel not to weaken the Lebanese government.

In Beirut, Hezbollah representatives said they wanted to ransom the Israeli captives for hundreds of Arabs imprisoned in the Jewish state, a swap akin to one agreed to by Olmert’s predecessor, Ariel Sharon, in January 2004.

But Olmert has ruled this out, describing the abductions by Hezbollah and Hamas in Gaza as part of a long-term Islamist war of attrition against Israel.

“We will not yield to extortion,” Olmert said Wednesday.

In his declaration, he “changed the world view espoused by Sharon, whereby Israel Defense Forces soldiers must be returned home even if it is at the steep price of a prisoner swap,” commentator Shimon Shiffer wrote in Yediot Achronot.

A test of stamina now looms. Many wonder how far Olmert is willing to push the campaign, or whether Lebanon, facing economic meltdown, will bend first.

“The Israelis have no other option but to swap. Hezbollah does not give in very easily,” Timur Goksel, a professor at the American University of Beirut who was the spokesman for the U.N. peacekeeping force in southern Lebanon for many years, told Reuters.

But he also criticized Hezbollah for triggering the fighting and said the militia’s political standing was at risk.

“The only conclusion I can reach is that they have decided to divorce themselves from Lebanese politics,” he said.