The Israeli prime minister said Mr Albanese's record had been ruined over the response to the conflict in Gaza, which included recognition of a Palestinian state.
"I'm sure he has a reputable record as a public servant, but I think his record is forever tarnished that he showed in the face of these Hamas terrorist monsters," he told Sky News.
The comments followed criticism from Mr Netanyahu on social media where he labelled Mr Albanese a "weak politician" and claimed he abandoned the nation's Jewish community.
The fracas followed the federal government denying a visa to a far-right Israeli politician set to undertake a speaking tour of Australia at the invitation of a fringe Jewish group.
Israel retaliated by cancelling the visas of three Australian representatives to the Palestinian Authority and ordering embassy staff to apply extra scrutiny to Australian applications to enter Israel.
Executive Council of Australian Jewry president Daniel Aghion wrote to both leaders to warn the deterioration in diplomatic relations brought pain and anguish, suggesting both men should have opted for silence.
"The Australian Jewish community will not be left to deal with the fallout of a spat between two leaders who are playing to their respective domestic audiences," he said.
Health Minister Mark Butler said the Israeli leader's comments were "frankly ridiculous".
"We're not going to focus on those (comments). We're going to focus on prosecuting a position that we think reflects Australia's national interest," he told Sky News.
"We've been very, very determined to make sure as far as possible that the division and the conflict that is happening over in the Middle East is not brought to our own country, that we focus on maintaining social cohesion."
A spokesman for Palestine's foreign ministry to Australia condemned Mr Netanyahu's comments towards Mr Albanese, as well as similar criticism of French President Emmanuel Macron.
"Conflating recognition of the State of Palestine with anti-Semitism and interfering in the internal affairs of France and Australia is unjustifiable, hostile to peace, and contrary to the international consensus on the principle of the two-state solution," the spokesman said.
Opposition Leader Sussan Ley urged the prime minister to repair diplomatic ties with Israel.
"Anthony Albanese has a mismanaged relationship, and that mismanagement clearly has consequences," she told reporters in Brisbane.
"What matters today is that we work hard together to get the relationship back on an even keel."
The tit-for-tat between Australia and Israel marked an all-time low in diplomatic relations but the impact remained to be seen, Australian National University politics and international relations senior lecturer Charles Miller said.
"Australia's championing of the rules-based global order looks hypocritical ... to the extent it is combined with close relations with Israel," he told AAP.
"By distancing itself from Israel, Australia gains credibility in many parts of the world, especially the Islamic world."
Israel has been accused of committing war crimes in its military action in Gaza, with the International Criminal Court issuing an arrest warrant for Mr Netanyahu.
The warrant, which includes allegations of using starvation as a method of warfare and directing attacks against civilians, are not findings of guilt and Israel denies the claims.
Australia could feel ramifications of the diplomatic low through a lack of intelligence communication with Israel, along with flow-on effects to its relationship with America, Dr Miller said.
But the US factor should not be exaggerated because it ought not be assumed America "will always remain as pro-Israel".
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