Under the hard-won 2015 deal with Russian Federation, the US, China, France, Britain, Germany and the EU, Iran agreed to curb its nuclear programme in exchange for lifting of a raft of worldwide sanctions.
Trump is also urging Congress to reform United States law so the president is no longer required to declare every 90 days whether he thinks Iran is in compliance, or to renew sanctions waivers every 120 or 180 days.
French President Emmanuel Macron called Trump on Thursday and stressed France's determination to see "the strict application of the deal and the importance of all the signatories to respect it".
The other signatories to the deal - Britain, China, France, Germany and Russian Federation - and the European Union, which oversaw the talks, were watching carefully.
Iran said on Saturday it would retaliate against the new sanctions, although it did not specify how. The Europeans, meanwhile, have said they are willing to discuss the matter with the US but have shown little enthusiasm with Trump's hard line.
"Trump's policy & today's announcement amount to desperate attempts to undermine a solid multilateral agreement, maliciously violating its paras 26, 28 & 29. Rather than repeating exhausted rhetoric, USA must bring itself into full compliance - just like Iran".
Trump has sharply criticized the agreement and vowed to rip it up, but his national security advisers have urged him to wait while they work with Congress and US allies in an effort to strengthen its provisions.
Trump will next have to deal with these decisions in mid-May.
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Most U.S. trade with and investment in Iran already is banned under separate U.S. sanctions for Tehran's human rights abuses, ballistic missile program and support for terrorism, but the administration and its supporters said more are needed. The Treasury Department's action hits 14 Iranian officials and companies and businessmen from Iran, China and Malaysia, freezing any assets they have in the USA and banning Americans from doing business with them. Under the current deal they are set to expire in 2025.
Trump said he would waive nuclear sanctions against Iran but only as a "last chance" and would not do so again.
"Instead, I have outlined two possible paths forward: either fix the deal's disastrous flaws, or the United States will withdraw". "No one should doubt my word".
Trump stopped short for a third time of re-imposing harsh sanctions meant to push Tehran to give up its nuclear weapons research.
Washington policy hawks hailed Trump's decision, which they said would concentrate minds in Europe, where leaders have urged the USA president not to sabotage a deal they see as a landmark diplomatic success.
United Nations inspectors have certified Iran's compliance with the deal nine times, most recently in November.
But the officials also want the triggers to include Iran's growing ballistic missile program, which was not part of the nuclear deal, and to remove sunset clauses that allow some nuclear restrictions to ease or phase out over time.