Iranian Foreign Minister Mohammad Javad Zarif had earlier said Iran would not break the terms of the agreement.
Mr Rouhani said Tehran will give 60 days for signatories to return to negotiating financial and oil deals.
This surgery is to save the deal, not destroy it.
Rouhani's address to the nation came on the anniversary of President Trump's decision to unilaterally withdraw the US from the landmark agreement.
Iran said the action was being taken because the European Union and others "did not have the power to resist United States pressure". The EU vowed to work on keeping legitimate trade with Iran, together with third countries interested in supporting the JCPOA, including through the special goal payment vehicle "INSTEX", which the EU set up earlier this year to keep trade payments-including oil trade-with Iran open.
Meanwhile, the ISNA news agency reported yesterday that Mr Zarif had travelled to Moscow to meet Russian leader Vladimir Putin to discuss bilateral and worldwide issues, without elaborating. That halted the promised worldwide business deals and dealt a heavy blow to Iran's already anemic economy. He did not elaborate.
US Secretary of State Mike Pompeo made an unannounced visit to Iran's neighbor Iraq on Tuesday to discuss the safety of Americans there, and explain US security concerns amid rising Iranian activity. World powers imposed crippling United Nations sanctions in response. A letter was also sent to Russian Federation.
Also reacting to the announcement, China called on all parties to uphold the Iranian nuclear pact.
Washington has also deployed an aircraft carrier strike group and several massive, nuclear-capable B-52 bombers to the Middle East as national security adviser John Bolton warned Washington would respond with "unrelenting force" to any attack by Tehran.
Iran reached the deal with world powers in 2015 after years of negotiations, including secret talks between Iran and President Barack Obama's administration in Oman. Iran has always maintained its program is for peaceful purposes.
Under terms of the deal, Iran can keep a stockpile of no more than 300kg of low-enriched uranium, compared with the 10,000kg of higher-enriched uranium it once had.
Under its policy of "maximum pressure", Washington has extended sweeping sanctions on Tehran and in recent weeks has hit even harder, moving to ban all countries from buying Iran's oil, its top export, and declaring the Revolutionary Guards to be a terrorist group - the first such designation of a unit of a foreign government.
On Friday, the U.S. acted to force Iran to stop producing low-enriched uranium and expanding its only nuclear power plant.
In a speech broadcast on national television, Rouhani said he had written to the remaining signatories saying Tehran would start rolling back on its commitments by stopping selling enriched uranium and heavy water to other nations.
However, once a country enriches uranium to around 20 per cent, scientists say the time needed to reach 90 per cent is halved. Iran has previously enriched to 20 percent.
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