In a televised speech, Mr Rouhani said Iran would"give a crushing response to America's bullying".
The Trump government said the measures were being reimposed under the mechanism of a deal that the US has stopped.
As a result, Iran began breaching some commitments it had made, including accentuating more energy than it should.
The US said the nation ought to be punished, announcing that all UN sanctions that were suspended would be reimposed - a mechanism agreed in the arrangement known as"snapback" - and also that a conventional arms embargo on Iran would no longer perish on 18 October.
Why do the limits on uranium enrichment matter?
How renewed US sanctions have struck Iran hard
But Washington's announcement has been disregarded by almost every member of the UN Security Council, and the body has not taken the measure any further. The effects of the conclusion remains unclear.
How has Iran responded?
In his speech, Mr Rouhani added:"America is coming a specific defeat in its sanctions movement... It faced defeat and a negative reaction from the global community."
Earlier, Iran's foreign ministry described the US attempts as"unworthy", stating that the"US approach is a major threat to the international peace and security and an unparalleled threat to the UN and the Security Council".
"Iran highlights that if the US, right or with the co-operation of a variety of its allies, makes any move consistent with these threats, it is going to face a significant reaction and should account for all its dangerous effects," it said in a statement, without elaborating.
What's the world reacted?
"It flows from this that any decisions and activities that are taken based on this process or on its possible outcome would also be incapable of having any legal effect," they stated in a joint announcement. "We have worked tirelessly to preserve the nuclear agreement and remain committed to do so."
The foreign ministry in Russia, another permanent member of the Security Council, said:"The illegitimate initiatives and activities of the USA by definition can't have international legal implications for different countries."
On Saturday, UN Secretary-General António Guterres told the Security Council he couldn't take any actions because"there would seem to be uncertainty" on the issue. However, US officials argue that, despite having abandoned the agreement, they nevertheless have the right to activate the"snapback" clause.
The statement was made 30 days after US Secretary of State Mike Pompeo informed the council that the administration was triggering the mechanism.
After abandoning the deal, the US unilaterally reinstated sanctions on Iran which, together with a drop in oil prices, have crippled the market in the nation, which has been hit hard by the Covid-19 pandemic.
The statement comes around six months before the US presidential election. Mr Trump faces Democratic rival Joe Biden who has pledged to bring the US back to the deal.
