Hezbollah's leader vowed Thursday to retaliate against Israeli interests anywhere in the world for the assassination of Imad Mughniyeh, one of the militant group's most notorious operatives, warning of a war without boundaries in a eulogy delivered to throngs of fist-waving mourners.

Israel ordered its military and embassies overseas to go on alert earlier in the day and recommended Jewish institutions around the world do the same, fearing revenge attacks for the car bomb that killed Mughniyeh Tuesday night in Damascus. The former Hezbollah security chief was one of world's most wanted fugitives, accused of masterminding attacks that killed hundreds of Americans and French in Lebanon in the 1980s.

The fiery speech by Hezbollah chief Hassan Nasrallah at Mughniyeh's funeral signaled that the Iranian-backed Shiite guerrilla group was ending a policy it has proclaimed for years of battling Israel only on Israeli or Lebanese territory, raising the specter of attacks in Western or other countries.

The day's events in Beirut also raised fears that Lebanon's internal turmoil could worsen. Earlier in the day, before Hezbollah supporters bid farewell to Mughniyeh, tens of thousands of their pro-Western political opponents filled a downtown Beirut square to mark former Prime Minister Rafik Hariri's 2005 assassination.

Fearing clashes, authorities deployed thousands of troops. The two mass gatherings ended without violence, but the two camps appeared even more deeply entrenched in their divisions, which have left the country without a president since November and the parliament paralyzed.

Hezbollah and its Iranian backers blamed Israel for killing Mughniyeh, but Israel denied involvement. In a videotaped eulogy broadcast on a giant screen to thousands attending the south Beirut funeral, Nasrallah said Israel had taken the fight outside the "natural battlefield" of Israel and Lebanon.

"You have crossed the borders," said Nasrallah, himself in hiding because of fears of assassination since the summer 2006 war between Israel and Hezbollah. "With this murder, its timing, location and method — Zionists, if you want this kind of open war, let the whole world listen: Let this war be open."

The thousands of black-clad mourners in the ceremony hall raised their fists in the air, chanting, "At your orders, Nasrallah."

In Washington, the State Department said it was not aware of a specific threat made to Israel by the Hezbollah leader but condemned anything of the sort as "alarming."

"As a general matter, those kinds of statements are quite concerning and they should be alarming to everyone," spokesman Sean McCormack told reporters. "Quite clearly, Hezbollah has a long record of carrying out violent acts and acts of terrorism around the globe. You have a pathway of violence that stretches from Buenos Aires to Kuwait and a lot of places in between."

Unlike many Middle Eastern leaders whose speeches are riddled with idle threats, Nasrallah is known for delivering on his promises. In 2006, he promised to take action to free Lebanese prisoners in Israel, and in July that year, Hezbollah guerrillas staged a daring cross-border raid that snatched two Israeli soldiers as bargaining chips.

The incident triggered a 34-day war between Israel and Hezbollah, devastating south Lebanon. It ended with the Israeli soldiers still captive, and no deal for a prisoner swap has been reached since.

Hezbollah has retaliated in the past for Israeli attacks. A 1992 car bombing of Israel's Embassy in Argentina came a month to the day after Hezbollah's secretary-general Abbas Mussawi was killed in an Israeli helicopter strike in Beirut. The 1994 bombing of a Jewish cultural center in Buenos Aries followed Israel's capture of a senior Hezbollah leader Mustafa Dirani and airstrikes on a Hezbollah training camp in Lebanon.

Western and Israeli intelligence agencies have accused Mugniyeh and Hezbollah of carrying out both those attacks, which killed more than 100 people. Mughniyeh was also believed linked to a 1996 bombing in Saudi Arabia that killed 17 Americans. Those were the last attacks outside Israel or Lebanon blamed on Hezbollah, though it denies responsibility.

The funeral hall in the Roueiss neighborhood of south Beirut was packed with mourners in front of Mughniyeh's coffin, draped in a Hezbollah flag. Two giant posters of the bearded, burly Mughniyeh in a cap and military fatigues were hung behind the coffin with a banner reading, "The Great Commander Martyr — Hajj Imad Mughniyeh." Some mourners cried as a band played Lebanon's national anthem and the guerrilla group's anthem. Outside in the rain, tens of thousands massed.

Nasrallah warned Israel that its alleged killing of Mughniyeh was a "very big folly" which will be avenged.

"Mughniyeh's blood will lead to the elimination of Israel. These words are not an emotional reaction," he said.

Iranian Foreign Minister Manouchehr Mottaki, who came to the funeral in Lebanon, offered condolences to the family and Mughniyeh's associates, before accepting condolences himself. Underlining Iran's close ties to Hezbollah, he sat between Mughniyeh's father and Hezbollah's deputy leader.

"He's not the first martyr, nor will he be the last on this path," Mottaki said, reading a statement of condolences from Iranian President Mahmoud Ahmadinejad with an interpreter translating into Arabic. "There will be hundreds and millions more" like him.

Afterward, the coffin was carried outside through the crowds of mourners, who marched with it to a nearby cemetery, praying aloud, as some chanted "Death to Israel" and "Death to America."

"We have lost a hero but this will not shake our will," said one Hezbollah guerrilla who would identifyd himself only by his first name, Hassan. The 45-year-old said Mughniyeh had been his commander in south Lebanon during the 2006 war. "We are all Imad Mughniyeh."

Another mourner, Hassan Awdeh, said Mughniyeh's death "only strengthens us and gives us zeal to complete our path."

"Imad Mughniyeh raised the head of our people high — from the 80's until his martyrdom," Awdeh's brother Ali added. "He's a model of the true Hezbollah man."

Mughniyeh's killing exacerbated tensions at a time when Lebanon is already entrenched in a long-running political crisis between the Hezbollah-led opposition and the Western-backed government. Government backers accuse Hezbollah of seeking to restore Syrian domination of the country, while the opposition says the government is putting Lebanon in the hands of the United States and Israel.

Earlier in the day, tens of thousands gathered in the main Martyrs' Square of downtown Beirut about 8 kilometers (5 miles) from the site of the funeral to commemorate the third anniversary of Hariri's assassination. The anti-Syrian rally appeared larger than the crowds at Mughniyeh's funeral, but it had been planned weeks in advance.

Waving Lebanese flags and carrying pictures of the slain Hariri, crowds also paid respects at Hariri's gravesite next to the downtown square as his brother, Shafik, unveiled a statue of the slain leader at the spot where he was killed, a few hundred yards away on a seaside boulevard. In the square, the sound of beating drums mixed with cheers from the crowd as speakers lashed out at the opposition.

Saad Hariri, leader of the parliamentary majority and the late premier's son, launched a scathing attack against the Syrian government. But he spared Hezbollah and its opposition allies, apparently in deference to the funeral. He even reached out to the opposition, saying: "Our hand will remain extended no matter what difficulties and conspiracies there are."

When Hariri alluded to Mughniyeh's funeral on the other side of the city, the crowd booed.

Druse leader Walid Jumblatt, a sharp critic of Hezbollah, said the government will not succumb to opposition efforts to deliver Lebanon "to the Iranian-Syrian black evil world." Jumblatt accused Syria and the "treacherous regime" of President Bashar Assad of killing Mughniyeh.

Hariri's supporters blame Syria for killing the prominent politician in a massive suicide truck bombing in Beirut and for a series of bombings and assassinations since. Hariri's assassination ignited mass protests and international pressure that forced Syria to withdraw its army from Lebanon after 29 years of control.

Amid fears of street violence, the U.S. Embassy encouraged Americans in Lebanon to limit all but essential travel Thursday. Across Beirut, businesses and shops put off popular Valentine's Day celebrations for later in the week. The Hariri rally ended just before the funeral ceremony began, and there were no frictions between the two camps.