On taking over from the assassinated Shaikh Ahmad Yasin, al-Rantisi vowed Israelis "will not know security" and threatened to "fight them everywhere".
He called on the military wing of Hamas, Izz al-Din al-Qassam Brigades, to "teach Israel a lesson".
Call to strike
"We tell the Izz al-Din al-Qassam Brigades: you must teach a lesson to the enemy. The door is open for you to strike all places, all the time and using all means," al-Rantisi said, addressing thousands of supporters.
"This enemy must come to the realisation that we fear nothing," he said.
Hamas supporters responded chanting: "We will sacrifice our blood and souls for you."
Senior Hamas official Ismail Haniya had earlier announced al-Rantisi had been chosen as the official deputy to Shaikh Yasin before his death.
"As a consequence, Dr al-Rantisi will take over from the shaikh as the head of the movement," he said.
Al-Rantisi confirmed his appointment, saying: "During its last meeting, the consultative council confirmed Shaikh Yasin as leader and elected me as deputy and, therefore, according to the statutes, the deputy replaces the leader if he is eliminated."
"We tell the Izz al-Din al-Qassam Brigades: you must teach a lesson to the enemy"
Abd Al-Aziz al-Rantisi
Khalid Mishaal, a Hamas leader based in Syria, still heads the group's political bureau, the main decision-making body.
A senior Hamas official said on condition of anonymity: "The successor to Shaikh Yasin is the internal leader while Khalid Mishaal is the head of the overseas political bureau."
Meanwhile, occupied Gaza Strip and the West Bank continued to mourn the death of Yasin.
Schools, universities and businesses are closed, reported Aljazeera’s correspondent.
Israel was on heightened alert for retaliatory attacks. Security has also been boosted at Israeli embassies around the world.