The Republican leader grudgingly agreed to sign sanctions waivers, ensuring Washington will live up to its commitments for another 120 days, but he cautioned it would be "for the last time."
During that four-month period, he now wants Congress and America's main European allies to draw up a new deal - without negotiating with Tehran - to replace the "disastrous flaws" in the current agreement.
"Despite my strong inclination, I have not yet withdrawn the United States from the Iran nuclear deal," Trump said in a statement.
"Instead, I have outlined two possible paths forward: either fix the deal's disastrous flaws, or the United States will withdraw."
The new deal - which Trump envisions as being outlined in US law and involving Britain, France and Germany - would impose tougher controls on Iran if sanctions relief is to continue.
In particular, it will not begin to expire after 10 years as parts of the existing deal do, but instead would impose permanent restrictions on not just Iran's nuclear plants but also its missile program.
"These provisions must have no expiration date. My policy is to deny Iran all paths to a nuclear weapon -- not just for ten years, but forever," Trump said.
"If Iran does not comply with any of these provisions, American nuclear sanctions would automatically resume."
Congress will also be urged to reform US law so Trump 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.
Iran's foreign minister, Mohammad Javad Zarif, took to Twitter to describe Trump's decision and the announcement as "desperate attempts to undermine a solid multilateral agreement."
The 2015 deal, he said, "is not renegotiable. Rather than repeating tired rhetoric, US must bring itself into full compliance -- just like Iran."
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Street protests
Trump's decision was announced alongside another package of new sanctions -- punishing not Iran's nuclear program, but its alleged abuse of human rights in a crackdown on recent street protests.
The most prominent new name on the Treasury sanctions blacklist, which forbids Americans from doing business with the targets, is the head of Iran's judiciary, Sadegh Amoli Larijani.
The US Treasury said Larijani, the brother of Iran's parliamentary speaker and former nuclear negotiator Ali Larijani, is to blame for the torture and degrading treatment of prisoners.
Washington policy hawks hailed Trump's decision, which they said would concentrate minds in Europe, where leaders have urged Trump not to sabotage a deal they see as a landmark diplomatic success.
"So Trump now has issued explicit May deadline to Congress and Europeans: Fix Iran nuke deal or no more sanctions waivers," wrote Mark Dubowitz, head of the Foundation for Defense of Democracies.
"Is he bluffing? I don't think so. Busy four months ahead."
But supporters of the existing deal between Iran and six world powers, including the former officials in Barack Obama's administration who negotiated it, denounced the plan.
Richard Nephew, a former US negotiator, tweeted that Trump had taken the US "down a very dangerous and unpredictable road."
In particular, he scorned the idea that a new arrangement could be negotiated without talking to the Iranians "who will have to carry out the results."
'Solid agreement'
There was no immediate response to the decision from the other signatories to the deal -- Britain, China, France, Germany and Russia -- nor the European Union, which oversaw the talks.
While there may be some relief that Trump has yet to torpedo the hard-won accord, there were clear signs of frustration from European capitals in the run-up to the decision.
America's allies see the accord as the best way to thwart Iran's nuclear ambitions and a victory for multilateral diplomacy. Tehran categorically denies it is seeking to develop atomic weapons.
But Trump argues that Obama gave away too much to Iran in sanctions relief, without forcing the Islamic republic to end its ballistic missile program and support for militant groups.
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."
And Russia spoke out strongly Friday in favor of the accord, calling it "the result of a consensus among many parties," in the words of Kremlin spokesman Dmitry Peskov.
In Brussels on Thursday, the European Union and the foreign ministers of Britain, Germany and France presented a united front after talks with Zarif.
"The deal is working, it is delivering on its main goal which means keeping the Iranian nuclear program in check," EU foreign policy chief Federica Mogherini said.
The agreement, she said, is "making the world safer and... preventing a potential nuclear arms race in the region."
UN inspectors have certified Iran's compliance with the deal nine times, most recently in November.
