The Indian government denied responsibility for the shootings, saying its forces had only used rubber bullets and tear gas to break up the crowd of 100,000 Kashmiri Muslims.
The protesters say they were demonstrating against road blocks errected by Hindus that have prevented them from reaching the Kashmir valley, damaging trade and blocking their access to vital goods.
Aziz was a prominent member of the All Parties Hurriyat Conference, an alliance of moderate Kashmiri separatist groups at the forefront of the political struggle against Indian rule in Kashmir.
"We will spill blood for blood," Aziz's supporters chanted as they carried his body out of the Srinagar hospital, prompting fears the killing could unleash a new wave of violence after several years of relative calm.
The deaths came as Indian security forces tried to prevent the Muslims from marching towards the de facto border with Pakistan.
The marchers had reached a point just 40km from the heavily militarised border, despite repeated efforts by Indian police and paramilitary forces to stop them with tear gas, rubber bullets and warning shots.
Police said the day's violence had also left at least 200 people injured.
Security forces imposed a strict curfew in Srinagar, the main city in Indian Kashmir and the hub of the 19-year-old revolt against New Delhi's control over the Muslim-majority region.
Kashmir has been hit by increased unrest in recent months, underscoring what locals say is resentment about the fact that the peace process between India and Pakistan, launched in 2004, has led to no progress on the issue that has caused two wars between the nuclear-armed neighbours.
The latest tensions stem from an order by the state government, issued in June, to donate land to a Hindu pilgrimage trust.
The decision sparked a series of violent protests by Muslims that left at least six people dead.
The plan was cancelled, but led to riots in Hindu-dominated Jammu, while Hindu hardliners then began blocking road access to the Kashmir valley, a move that has badly affected Muslim traders.
The blockade has led to shortages of essentials such as medicines, which prompted the protest march to Pakistan, so that fruit growers and traders could sell their produce on the other side of the border.
"This is the real face of Indian democracy," said Mirwaiz Umar Farooq, the head of the moderate separatist alliance, adding that Aziz's death was a "big loss" to the separatist movement.
"For demanding lifting of an economic blockade we get bullets, while Hindu fanatics who are attacking Muslims in Jammu and setting their property on fire are allowed to do whatever they want," he said.
Aziz, who had been jailed on several occasions for demanding Indian Kashmir be handed to Pakistan, is the third prominent separatist leader to have been killed since the eruption of the Muslim revolt in 1989.
Fresh tensions were also reported along the Line of Control on Monday, with the Indian army accusing Pakistan of another ceasefire violation that sparked a brief exchange of mortar and small-arms fire.
New Delhi accuses Islamabad of pushing militants into the Indian part of Kashmir, a charge Pakistan denies.