Americans are dreaming if they think they can visit Iran's military sites, says Khamenei adviser
"Our commitments to the world are clear... our relations with the IAEA (International Atomic Energy Association) are defined by rules, not by the US," Rouhani said in a televised address.
"I see it as unlikely that the IAEA will accept the request for inspections, but even if they do, we will not," he said.
The comments came in response to media reports claiming Washington's UN envoy Nikki Haley discussed accessing Iranian military sites in order to ensure Iran's compliance with the 2015 JCPOA deal, during a meeting with IAEA chief Yukiya Amano last week.
The landmark accord saw international sanctions eased in exchange for stringent controls on Iran's nuclear programme and closer IAEA inspections.
The inspections are designed to prevent nuclear material being moved from nuclear centres to other areas, including military bases.
But the IAEA has repeatedly reported that Iran is in compliance with the deal.
Rouhani said Iran had the support of Europe and other signatories, and added that US President Donald Trump would struggle to muster support for tearing it up.
"The US is in the most difficult situation in its history for creating unity against us and I think it's not possible for it to do so now," Rouhani said.
"Twenty-eight EU countries, which are America's allies, clearly say we are committed to the JCPOA," he added.
Earlier on Tuesday, French President Emmanuel Macron confirmed his commitment to the deal and said there was "no alternative".
Ali Akbar Velayati, supreme leader Ayatollah Ali Khamenei's foreign policy adviser, also dismissed the reported demand for inspections as "sloganeering by the new US administration for domestic consumption".
"The Americans should take the dream of visiting our military sites, using the pretext of the JCPOA or any other pretext, to their graves," he said.
Despite the pushback, the deal remains under threat.
The US has strongly criticised recent Iranian missile tests and imposed new, non-nuclear sanctions – prompting Rouhani to hint in mid-August that Iran could easily walk away from the deal "within hours" if forced to do so.
Haley responded that new US sanctions against Iran relate to its support for "worldwide terrorism" and other destabilising actions in the region.
Tehran cannot "use the nuclear deal to hold the world hostage," the US official said.
Trump is due to notify Congress in October about whether Iran is adhering to the deal.