The comments from Defence Minister Israel Katz came after the Soroka military Hospital in the southern city of Beersheba reported 40 people injured after an Iranian missile strike.
"Khamenei openly declares that he wants Israel destroyed -- he gives the order to fire on hospitals," Katz told journalists in Holon near Tel Aviv.
"Such a man can no longer be allowed to exist."
Iran said the main target of the attack was an Israeli military and intelligence base, beneath the health facility.
"The main target of the attack was the Israeli Army Command and Intelligence Base (IDF C4I) and the Army Intelligence Camp in Gav-Yam Technology Park, located in the vicinity of the Soroka Hospital," state news agency IRNA said.
It said the hospital was "exposed only to the blast wave" and that the military facility was the "direct and precise target."
A senior US official told AFP on Sunday that President Donald Trump had "found out that the Israelis had plans to hit Iran's supreme leader".
"President Trump was against it, and we told the Israelis not to," said the US official, speaking anonymously.
In a television interview on Monday, Israeli Prime Minister Benjamin Netanyahu himself did not rule it out, saying that killing the 86-year-old cleric who has ruled Iran since 1989 would "end the conflict" between the two countries.
Trump wrote on Tuesday that the United States knew Khamenei's location but would not kill him "for now".
Israel has a long-established history of political assassinations in the region. Over the past decades, it has carried out targeted assassinations of political leaders in Palestine, Iran, Syria, and Lebanon.
It launched strikes on Iran last Friday in what it claims was a move to prevent it from acquiring nuclear weapons.
It has since hit hundreds of targets, including military commanders, top nuclear scientists, military and nuclear facilities, and residential buildings.
'Regime change'
Netanyahu has not said publicly that Israel is trying to topple him, only that regime change could be a result of its military action.
Iranians "understand that the regime is much weaker than they thought -- they realise it, and that could lead to results," he told a press conference on Monday.
French President Emmanuel Macron has said that any attempt at forcing change through military action would result in "chaos", while both China and Russia demanded that Israel cease fire.
Iran denies seeking to develop a nuclear weapon, and reports citing US intelligence officials this week have cast doubt on Israeli claims that it has accelerated efforts to produce one.
Iran has been enriching uranium to 60 per cent -- far above the 3.67-percent limit set in a 2015 nuclear deal, which Trump abandoned, but still short of the 90-percent threshold needed for a nuclear warhead.
Israel has maintained ambiguity on its own nuclear arsenal, but the Stockholm International Peace Research Institute says it has 90 nuclear warheads.
The International Atomic Energy Agency (IAEA) accused Iran of a lack of cooperation just before the start of the Israeli attacks. The IAEA's board of governors then adopted a resolution censuring Iran for "non-compliance" with its obligations under the Nuclear Non-Proliferation Treaty.
However, the head of the agency, Rafael Grossi, had said, "we did not have any evidence of a systematic effort [by Iran] to move into a nuclear weapon.”
Israel has its own secretive nuclear weapons program, one that it doesn’t publicly acknowledge but that, experts told the New York Times, is also expanding.
Israel is widely believed to have at least 90 warheads and enough fissile material to produce up to hundreds more, according to the Center for Arms Control and Nonproliferation and the Nuclear Threat Initiative.
According to the center, Israel could fire warheads from fighter jets, submarines or ballistic missile ground launchers.
Israel is one of five countries — joining India, Pakistan, North Korea and South Sudan — that is not a signatory to the UN Nuclear Nonproliferation Treaty.
The agreement, which came into force in 1970, generally commits governments to promoting peaceful uses of nuclear energy and preventing the spread of nuclear weapons. Israel would have to give up its nuclear weapons to sign the treaty, which recognizes only five countries as official nuclear states: Britain, China, France, Russia and the United States — the permanent members of the UN Security Council.
* This story was edited by Ahram Online.
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