Senior Russian figures, including the head of the military, have warned that USA missiles will be shot down and their launch sites targeted if Russian personnel come under threat.
Ambassador Antonov said: "Insulting the President of Russian Federation is unacceptable and inadmissible".
In a statement, Australian Prime Minister Malcolm Turnbull said Australia supported the strikes, which sent "an unequivocal message to the Assad regime and its backers, Russian Federation and Iran, that the use of chemical weapons will not be tolerated".
The Kremlin said Mr Putin warned against rushing to blame the Syrian government before conducting a "thorough and objective probe".
The meeting was called by Russian Federation to address USA threatened military action in response to last weekend's attack in the Damascus suburb of Douma. There is no evidence for now that the Syrian government has used chemical weapons, Azimov said.
Haley spoke directly after Russian Ambassador to the UN Vassily Nebenzia, who charged that the Saturday attack on Douma, the last rebel-held town in Syria, was faked by foreign intelligence services.
Another highly placed Russian politician likened President Trump to Adolf Hitler and said the air assault is regarded as a move against Russia. Both the Russian military and the Syrian government said they would facilitate the mission and ensure the inspectors' security.
As fears of a Russia confrontation with Western powers mount, French President Emmanuel Macron expressed his "deep concerns" over the situation in Syria in a phone call with Russian President Vladimir Putin.
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The chairman of the worldwide affairs committee of Russia's upper house of parliament, Konstantin Kosachev, said the strikes were a violation of worldwide law and probably created to prevent investigators from the global chemical weapons watchdog from doing their work. Moscow alleged soon after the suspected April 7 attack that the images of the victims in Douma were fakes.
She noted that In 2017, Moscow "killed" a Joint Investigative Mechanism, which the United Nations established in 2015 to investigate and assign accountability for chemical weapons use in Syria.
Yildirim on Friday again condemned the "heinous" the attack in Douma and said the perpetrators should "pay a price".
Syrian opposition activists and medics say a suspected gas attack last week killed more than 40 people in Douma.
The Qatari news site Al Arabiya reported a huge blast heard near Damacus Friday night, citing several media outlets.
At the same time, we will support the regime of Bashar al-Assad militarily more than we have so far.
"Russia in the most serious way condemns the attack on Syria, where Russian military servicemen help the legitimate government to fight terrorism", Putin said.