Rouhani's remarks came in response to a speech by Trump earlier on Friday in which the US president said he would not continue to certify the landmark deal, also signed in 2015 by Britain, France, Russia, Germany and the European Union.
Speaking from Tehran, Rouhani said Trump's speech was "full of insults and fake accusations" against Iran.
He called the deal not renegotiable, adding that Trump's decision to decertify it would isolate the US as other signatories of the accord remained committed to it.
"Iran and the deal are stronger than ever," Rouhani said in a televised address.
In his speech, Trump accused Iran of malign behaviour and announced new sanctions against its elite military wing, the Islamic Revolutionary Guards Corps (IRGC). The US president also called on allies to help impose sanctions that would target Tehran's ballistic missile programme.
In a defiant response, Rouhani said Iran's ballistic missile programme will expand despite pressure from the US.
"The Iranian nation has not and will never bow to any foreign pressure," said Rouhani, adding that the IRGC "will continue its fight against regional terrorists".
As reporting Al Jazeera, Rouhani's speech as "very strong", with a "tone that was meant to match the aggressive" rhetoric coming from the White House.
Iran's foreign ministry also issued a statement in response to Trump's speech, citing the repeated confirmations by the IAEA, the UN nuclear watchdog, that Tehran was complying with its obligations under the deal.
"The claim by the US president regarding Iran's non-compliance has no international relevance or credibility, since the IAEA, as the sole authority to conduct such monitoring and verification, has repeatedly confirmed Iran's full compliance with its commitments," the ministry said.
"This proves once again that the United States is not a reliable negotiating partner," he added.