|Pillay said 'Israel has a duty to ensure that its security personnel avoid the use of excessive force' [EPA]
The UN's human rights chief has said she was "deeply troubled" by the deadly unrest in the Golan Heights, where up to 40 people have reportedly been killed in recent protests.
"Between 30 and 40 protestors have reportedly been killed by Israeli security forces in the past three weeks," Navi Pillay said in a statement on Tuesday.
"The government of Israel has a duty to ensure that its security personnel avoid the use of excessive force," said the UN High Commissioner for Human Rights.
"However difficult the circumstances, the use of live ammunition against allegedly unarmed protestors, resulting in large numbers of deaths and injuries, inevitably raises the question of unnecessary and excessive use of force."
Israeli security forces used live fire against civilians protesting along the ceasefire line between the Golan Heights and Syria on Sunday.
The rally on "Naksa Day" (Day of Defeat) was to mark the Arab defeat in the 1967 war against Israel.
Protesters there were calling for an end to the occupation of the Golan.
Syrian state television said 23 demonstrators were killed by Israeli forces during the rally at the Golan Heights ceasefire line, while Israel's military said it counted 10 protesters dead - none of whom was killed by Israeli fire.
In her statement on Tuesday, the UN's Pillay also called on both sides to carry out independent, impartial, transparent and thorough investigations into the events of 5 June.
She reminded Israel of its obligation to carry out investigations into the events surrounding the protests of 15 May in the occupied Palestinian territory, the occupied Golan Heights and on the Lebanese border, which resulted in the loss of 15 civilian lives.
Meanwhile, in the Palestinian Gaza Strip, a few thousands of Hamas supporters participated in a rally to honour the Syrian protesters killed on Sunday. Demonstrators carried Hamas flags in addition to banners praising the "martyrs" of the Naksa Day.
A Syrian government newspaper said more Syrians and Palestinians plan to march to the Israeli border and that Israel should expect a march "at any time".
Tishreen said the march was only an "introduction'' adding that Syrians and Palestinians were now determined to recover their territory through resistance.
It said Israel should not be surprised when 600,000 Syrian refugees march back to their villages and farms from which their families were forcefully uprooted.
In Washington, state department spokesman Mark Toner said the US believes President Bashar al-Assad's government is actively supporting the protests at the border.
"We don't have any hard evidence," Toner said. "But we've seen this kind of behaviour before. And certainly it seems in keeping with the Syrian regime's actions that they would try to deflect or distract international
attention from what's going on internally in Syria by encouraging these kind of protests.''
On Monday, a funeral for activists shot dead by Israeli forces turned into violent protests against a Palestinian faction in a Syrian refugee camp, witnesses and officials said.
The violence broke out at the Yarmouk Palestinian refugee camp in Damascus, a day after the shootings by Israeli forces.
Mourners at Yarmouk accused the Syrian-backed Popular Front for the Liberation of Palestine-General Command (PFLP-GC), the group organising Sunday's protest, of inciting young people to put themselves in the firing line.
A witness told the AFP news agency that gunfire broke out while some mourners were at the cemetery.
He said that the crowd at Yarmouk was chanting slogans against the leaders of the group, a splinter faction from the Popular Front for the Liberation of Palestine (PFLP), and that some tried to damage vehicles belonging to it.
The Reuters news agency reported that armed men from the PFLP-GC shot dead at least 11 Palestinians at the refugee camp after they they tried to storm the group's headquarters.