The comments late on Friday came after the USA called an emergency meeting at the UN Security Council to address the deadly demonstrations that began in late December in Iran.
British Ambassador Matthew Rycroft concurred with Haley.
"The Iranian regime is now on notice".
"Tomorrow the U.N. Security Council will discuss the troubling and risky situation in Iran", Haley said in a statement issued Thursday evening, adding that it was "a matter of fundamental human rights for the Iranian people, but it is also a matter of global peace and security".
But Moscow stopped short of seeking to block the meeting. Reporting in Tehran, Al Jazeera's Zein Basravi said the subsequent pro-government rallies were meant to show Iran's rivals - both inside and outside the country - that there is considerable support for its political establishment. There were some reports of small anti-government demonstrations in provincial towns, but these could not be verified.
Several countries on the Security Council have blocked or attempted to prevent meetings focused on internal or domestic affairs of United Nations member states in the recent past.
"Let Iran deal with its own problems", Nebenzia advised, adding, "The real reason for convening today is not to protect human rights or promote the interests of the Iranian people, but rather a veiled attempt to continue to undermine the Iranian nuclear agreement", cited by the New York Times. And what might happen if US President Donald Trump decides on 11 January that he will no longer waive the sanctions contributing to Iran's stagnant economy?
Washington has imposed unilateral sanctions on five Iranian companies linked to Tehran's ballistic missile program.
"The parliament is not in favour of keeping Telegram filtering in place, but it must pledge that it will not be used as a tool by the enemies of the Iranian people", Behrouz Nemati, spokesman for the parliament's presiding board, said.
The unrest has left at least 21 dead, and is the biggest such movement since the protests in 2009 over a disputed election.
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.
Britain and France reiterated that Iran must respect the rights of protesters, but French Ambassador Francois Delattre said the "events of the past days do not constitute a threat to peace and worldwide security".
China's Deputy Ambassador Wu Haitao in his remarks said situation of Iran at the council "does not help resolve the domestic issue of Iran".
His administration has also cast doubt on the landmark 2015 deal on Iran's nuclear activities. "Another FP blunder for the Trump administration", Zarif tweeted on Friday.