Anti-government protests have swept Iranian cities and towns over the past week and 21 people have been killed in the unrest.
In a rebuke to Iranian government claims that the widespread demonstrations had been organized and/or instigated by foreigners, a group of 16 prominent reformist figures living in Iran issued a statement rejecting that claim.
Russia's envoy shot back that if the US view holds, the council should have also discussed the 2014 unrest in Ferguson, Mo., over the police shooting of a black teenager or the USA crackdown on the Occupy Wall Street movement. "The world will be watching what you do", Haley told a meeting of the United Nations Security Council.
United States President Donald Trump has come out in support of the protestors.
In order for the Council meeting that Haley wants to take place, she would need a minimum of nine votes from Council members.
But even before the session began, France's ambassador, Mr François Delattre, warned against "instrumentalisation" of the protests "from the outside".
"We must be wary of any attempts to exploit this crisis for personal ends, which would have the diametrically opposed outcome to that which is wished".
Member nations agreed with the Russian ambassador's assessment. He asked rhetorically why the Security Council had not taken up the issue of Black Lives Matter protests in Ferguson, Missouri, which were at times also met with a violent police response.
"The real reason for convening today is not protect human rights or promote the interests of the Iranian people, but rather a veiled attempt to continue to undermine the Iranian nuclear agreement".More news: NFL Wild Card Preview: Falcons at Rams
Mr Trump has repeatedly excoriated the deal, which was a signature diplomatic achievement of his predecessor, Mr Barack Obama.
Under the deal, Trump must actively lift certain sanctions every few months and the next deadline falls on January 12.
According to Al-Jazeera, Nebenzia said if the UN Security Council followed the reasoning behind convening the meeting over domestic protests in Iran, it should have met after demonstrations in the US following the killing of African-American teenager Michael Brown by a white police officer in the state of Missouri in 2014.
At the Security Council on Friday, most members insisted that these two issues were separate.
"This is how we view the American initiative to convene the UN Security Council over a situation that is of purely national concern in Iran".
Such reticence to support the USA position is the latest evidence of growing global resistance to the Trump administration's foreign policy priorities, particularly at the United Nations.
The United States and Israel have expressed support for the protesters, but deny allegations of fomenting them.
She called on all of her other colleagues in the Security Council to join her in amplifying the message of the Iranian people.