Nikki Haley also said the worldwide community can't let Iran silence protesters' messages, particularly by blocking some social media platforms.
Earlier in the week, the US government announced it was suspending security assistance to the country.
Russia's Permanent Representative Vassily Nebenzia said that if the Council followed the reasoning behind convening the the meeting over domestic protests in Iran, it should have met after the protests in the USA that followed the killing of an African-American teenager in Ferguson by a white police officer in 2014.
At least 21 people have been killed amid the anti-government protests and unrest over the country's economic woes - demonstrations Iran has accused the US of stirring up. But tens of thousands have also reportedly taken part in pro-government rallies in recent days. A similar rally followed Friday prayers in Tehran.
Solidarity with the historic resistance of the people of Iran for freedom
Hundreds of people took to the streets of several European cities, Canada and the United States expressing solidarity with the Iranian uprising and calling for the Khamenei regime to step down.
The meeting was a defeat for USA diplomatic interests as countries in the Security Council defended the principles of sovereignty and non-interventionism.
Echoing Iran's concern that the demonstrations were fueled by outside powers to promote regime change, Bolivia reminded world powers that "unilateral action, political interventionism, and regime change only ever lead to devastating humanitarian, social and economic consequences".
But Iran's Ambassador to the UN Gholamali Khoshroo criticized the council for taking up a matter his country considers "purely domestic", and called it bullying by the United States.More news: Panthers Wild Card Playoffs preview and predictions
The statement drew vehement retorts from Russian Federation and Iran, which drew attention to protests within the USA that had been put down.
Haley said "the people of Iran are rising up" and "if the founding principles of this institution mean anything, we will not only hear their cry, we will finally answer it". "Let Iran deal with its own problems".
Envoys from several other countries including China expressed reservations about whether the council was the right place to debate the issue.
Under its charter, the United Nations empowers the council to "investigate any dispute, or any situation which might lead to global friction".More news: Falcons Hang On Against the Rams for a Tough Playoff Win
Hundreds carried out demonstrations in front of the Brandenburg Gate in Germany in support of Iranian protesters facing repression from the government in the past two weeks. The country has increasingly has blamed its foreign foes for fomenting the unrest.
"We will not be quiet", she told the meeting.More news: Fodder scam: RJD Chief Lalu Prasad sentenced to 3.5 years in jail