It comes amid a surge in violence between Israel and the Palestinians.
The Palestinian president said he held the Israeli government fully responsible for what he described as a "crime of execution".
IMAGE SOURCE, EPA
Shireen Abu Aqla was known to millions for her coverage of the Israeli-Palestinian conflict
Shireen Abu Aqla, 51, was widely known and widely admired - by viewers and colleagues alike.
Early on Wednesday morning she went to the Jenin refugee camp to report on a raid by Israeli soldiers and security forces, which the Israeli military said was conducted to apprehend "terrorist suspects".
"During the activity, tens of Palestinian gunmen fired at and hurled explosive devices toward the soldiers. The soldiers responded with fire toward the gunmen and hits were identified," a military statement said.
The Palestinian health ministry said Abu Aqla was hit in the head by a live bullet during the raid. She was taken to hospital in a critical condition and later pronounced dead.
Another Palestinian journalist, Al Jazeera producer Ali Samoudi, was shot in the back and was in a stable condition in hospital, the health ministry added.
"We were going to film the Israeli army operation and suddenly they shot us without asking us to leave or stop filming," Al Jazeera cited Mr Samoudi as saying. "The first bullet hit me and the second bullet hit Shireen.
"There was no Palestinian military resistance at all at the scene," he added.
Video of the shooting showed Abu Aqla was wearing a blue flak jacket clearly marked with the word "press", as well as a helmet.
Shireen's body has been carried through the streets of Jenin, draped in a Palestinian flag and covered with her press flak jacket. Reporters and photographers packed in around the procession - having to cover the death of one of their own.
Shireen was a well loved reporter, one of the region's most experienced; always a familiar face at the many big news events that break here. A generation of Palestinians grew up seeing her on their TV screens, one of the best known women reporters covering the conflict.
In a tight-knit press corps, there is shock and disbelief. One colleague posted what he said was the last picture of her with them, putting on her flak jacket five minutes before her death. Another said of Shireen: "She saw death so many times during her coverage. You taught us how to do this dangerous work. You just broke our hearts."
A statement from Al Jazeera said: "In a blatant murder, violating international laws and norms, the Israeli Occupation Forces assassinated in cold blood Al Jazeera's correspondent in Palestine, Shireen Abu Aqla, targeting her with live fire early this morning... while conducting her journalistic duty."
The network called on the international community to hold the Israeli government and military accountable for the "intentional targeting and killing" of a journalist.
Qatar - which funds Al Jazeera - said it considered the killing a "heinous crime and a flagrant violation of international humanitarian law and a blatant infringement on freedom of media and expression".
Palestinian Authority President Mahmoud Abbas condemned the shooting of Abu Aqla and Samoudi and alleged that it was "part of the occupation's policy of targeting journalists to obscure the truth and commit crimes silently".
IMAGE SOURCE, REUTERS
Al Jazeera producer Ali Samoudi, who was wounded, said the fire came from Israeli soldiers
Israeli Prime Minister Naftali Bennett accused Mr Abbas of making "baseless accusations".
"According to the information we have gathered, it appears likely that armed Palestinians - who were firing indiscriminately at the time - were responsible for the unfortunate death of the journalist," he said in a statement.
"Palestinians in Jenin were even filmed boasting: 'We hit a soldier; he's lying on the ground.' However, no [Israeli] soldier was injured, which increases the possibility that Palestinian terrorists were the ones who shot the journalist."
Israel's foreign ministry and the Israeli military tweeted a video showing a Palestinian gunman firing down an alleyway in the Jenin camp.
In response, the Israeli foreign ministry stressed that "there was no claim that the gunfire in the clip killed" Abu Aqla.
Israeli military chief Lt Gen Aviv Kohavi meanwhile issued a statement saying: "At this stage, it is not possible to determine whose shooting she was hit by and we are sorry for her death."
Mr Bennett also said Israel had called on the Palestinian Authority to conduct a joint post mortem and investigation in order to get to the truth. He claimed that Palestinian officials had so far refused the offer, but a Palestinian minister said there had been no contact from Israel about a joint probe.
The US ambassador to Israel, Tom Nides, said he was "very sad" to learn of Abu Aqla's death and that he encouraged a thorough investigation.
UN envoy Tor Wennesland called for those responsible to be held accountable. "Media workers should never be targeted," he said.
Abu Aqla joined Al Jazeera in 1997 and was one of its first field correspondents. She was one of the best known women to become a regular face on TV screens covering the Israeli-Palestinian conflict over the years.
Her colleague Nida Ibrahim said she was "very well respected" and that the news was "a shock to the journalists who have been working with her".
"She was a household [name] in Palestine, in the region and worldwide for being who she was - a prominent journalist who spent her life covering news, covering Israeli brutality in Palestine," he said.
IMAGE SOURCE, REUTERS
Abu Aqla's death prompted outpourings of grief from journalists who knew her
The Israeli military frequently launches arrest raids into Jenin refugee camp. They have stepped up recently following a wave of attacks by Israeli Arabs and Palestinians on the streets of Israel and the West Bank in recent weeks that has left 17 Israelis and two Ukrainians dead.
At least 26 Palestinians have been killed - including assailants shot dead while carrying out attacks, or militants and civilians killed during Israeli raids and confrontations in the West Bank.
Israeli operations have centred on the Jenin district, where four of the Palestinians who carried out attacks in Israel came from.