The Iranian foreign ministry has said it would not accept “any changes, now or in the future”.
Foreign Minister Javad Zarif added that the demands were a “desperate attempt” to undermine the deal.
Mr Trump was the one that needed to check his compliance, Mr Zarif added.
President Trump made a temporary renewal of the agreement, which was negotiated by his predecessor Barack Obama‘s administration.
He had previously described it as “the worst deal ever”.
However, he said it would be the final time that economic sanctions would be waived by the US, although the deal’s European signatories – the UK, France, and Germany, alongside Russia and China – reaffirmed their commitments.
Following the announcement, Mr Zarif tweeted: “Trump’s policy & today’s announcement amount to desperate attempts to undermine a solid multilateral agreement, maliciously violating its paras 26, 28 & 29.
“The Joint Comprehensive Plan of Action (JCPOA) is not renegotiable: rather than repeating tired rhetoric, the US must bring itself into full compliance – just like Iran.”
Iran agreed to restrict its nuclear programme for at least 10 years in exchange for the relaxation of sanctions which have hamstrung its economic development in recent years.
Despite his objections this is the third time President Trump has renewed the deal and UK Foreign Secretary Boris Johnson said on Friday “nobody has so far produced a better alternative”.
Mr Trump maintains that Iran is not complying with the spirit of the deal, despite his own advisors previously admitting there had been no technical violations.
“Either fix the deal’s disastrous flaws, or the United States will withdraw,” he said.
The US President is demanding Iran allow immediate inspections at nuclear sites on request, and that so-called “sunset clauses”, which see the restrictions on nuclear development relax after 10 years, be removed.
“My policy is to deny Iran all paths to a nuclear weapon – not just for ten years, but forever,” Mr Trump said.
He also wanted to extend legislation to make explicit that testing of long-range missiles was “inseparable” from the nuclear weapons programme, so Iran’s ballistics testing should also “be subject to severe sanctions”.
State-run news outlet, IRNA, reports that Iran’s foreign ministry responded to the demands, saying “Iran strongly announces that it will make no measure beyond its JCPOA commitments and will make no changes in the nuclear deal neither now nor in the future”.
Even though Mr Trump has now imposed a 120-day deadline on his European partners, it does not appear like they have the appetite to negotiate a follow-on agreement regarding Iran.
Earlier this week European foreign ministers met in Brussels with Mr Zarif, seemingly to press Tehran about its destabilising activities in the Middle East.
“The Iran nuclear deal makes the world safer. European partners were unanimous today in our determination to preserve the deal and tackle Iran’s disruptive behaviour,” added Mr Johnson.
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies