The new local leader of Palestinian militant group Hamas has come to power with a call to arms, urging followers to unite in teaching Israel "a lesson".
Mr Rantissi is regarded as a hardliner and a firebrand orator
Israelis "will know no security", said Abdel Aziz al-Rantissi, the man named on Tuesday as Hamas' Gaza chief.
He was addressing crowds in Gaza that had gathered to mourn Sheikh Ahmed Yassin, the elderly Hamas founder killed by Israeli missiles on Monday.
Earlier on Wednesday Israeli tanks entered the southern Gaza Strip.
Israel's security forces are on high alert in expectation of a Hamas strike.
Its borders with Palestinian areas in the West Bank have been sealed and security checks have been stepped up.
Local residents said that Israeli tanks entered a refugee camp near Khan Younis early on Wednesday.
Bulldozers destroyed several houses opposite a Jewish settlement, forcing about 60 families to flee.
There were exchanges of fire reported between Israeli soldiers and Palestinian gunmen, but there were no reported of casualties.
Earlier, Israeli naval gunships were reported to have fired at targets in the Gaza Strip.
And Israeli helicopter gunships struck at what Israel called a band of guerrillas in south Lebanon at the same time, killing at least one.
Israel denied a report that the guerrillas had fired rockets across the border, but said the fighters had been preparing an attack on Israel.
Israeli government spokesman Avi Pazner denounced Mr Rantissi's appointment, describing him as "one of the most extreme members of Hamas".
The Israeli army's chief of staff Moshe Yaalon earlier said he expected Sheikh Yassin's killing to strengthen Palestinian moderates in the long run.
Troops tighten security in Israel's major cities, fearing attack
Questioned over the killing of Sheikh Yassin, US President George Bush said on Tuesday that Israel had a "right to defend itself from terror" - but should also bear in mind the consequences.
President Bush re-iterated his support for a two-state solution to the Israeli-Palestinian dispute and for a concerted fight against terror.
On Tuesday evening, the United Nations Security Council began a debate after its 15 members failed to reach unanimity on a statement about the killing of Sheikh Yassin.
The BBC's UN correspondent says the Palestinians and their allies have drawn up a draft resolution condemning the killing but this has not been officially tabled.
The United States says it won't support any resolution which does not also criticise Palestinian violence.
The Palestinian observer to the UN, Nasser al-Kidwa, said he strongly condemned what he called a war crime.
But the Israeli ambassador to the UN Dan Gillerman replied that Sheikh Yassin was an arch terrorist and one of the greatest obstacles on the road map to peace.
Announcing he had been elected to head Hamas in Gaza, Mr Rantissi exhorted the Islamist movement's military brigade to avenge Sheikh Yassin's killing.
"The door is open for you to strike all places, all the time and using all means," Mr Rantissi said.
Crowds at Gaza's soccer stadium reacted to the words of the Hamas hardliner with cries announcing their readiness to be martyred for the cause.
Mr Rantissi, a 56-year-old doctor who has been imprisoned by both Israel and by Yasser Arafat's Palestinian Authority, earlier told the BBC that no-one could replace Sheikh Yassin.
He added that the group's main leader was Khaled Meshaal, a Damascus-based academic who oversees Hamas' political bureau.
In 2003, Dr Rantissi survived an Israeli assassination attempt similar to that which killed Sheikh Yassin.