He described it as an "embarrassment" to the US that does nothing to contain Iran's nuclear ambitions.
In response, the foreign ministers of France, Britain and Germany will meet with Iranian representatives next Monday "to consider the entire situation," France's Foreign Minister Jean-Yves Le Drian told RTL radio.
French President Emmanuel Macron will also speak with his Iranian counterpart Hassan Rouhani by telephone on Wednesday afternoon about "our wish to stay in the agreement," Le Drian said.
He added that European powers would "try to preserve" the economic benefits Iran has gained from the lifting of sanctions under the deal.
In separate comments, French Economy Minister Bruno Le Maire said that it was "not acceptable" for the US to be the "economic policeman of the planet".
Slapping aside more than a decade and a half of diplomacy by Britain, China, France, Germany, Iran, Russia and past US administrations, President Trump called for a "new and lasting deal".
He said that would have to include not just deeper restrictions on Iran's nuclear program but on its ballistic missiles and support for militant groups across the Middle East.
"We cannot prevent an Iranian nuclear bomb under the decaying and rotten structure of the current agreement," he claimed.
"We will not allow American cities to be threatened with destruction and we will not allow a regime that chants 'Death to America' to gain access to the most deadly weapons on Earth."
Iran's leader Rouhani - whose standing at home now risks being undermined by the deal's collapse - was furious, accusing Trump of "psychological warfare".
Rouhani said Iran could resume uranium enrichment "without limit" in response to President Trump's announcement, but that it would discuss its response with other parties to the deal before announcing a decision.
President Trump's demands and his warning that Iranians deserve better than their current "dictatorship" will fuel suspicions that his ultimate goal is regime change.
"If the regime continues its nuclear aspirations, it will have bigger problems than it has ever had before," he warned.
Iran's Supreme Leader Ayatollah Ali Khamenei said Trump's comments on withdrawing were "silly and superficial".
"I said many times from the first day: don't trust America," Khamenei said. And he added, "I don't trust these three countries," Britain, France and Germany.
Blow for Europe, Obama speaks out
The decision marked a stark diplomatic defeat for Europe, whose leaders begged the US leader to think again.
In a joint statement, Germany's Angela Merkel, Britain's Theresa May and France's Macron voiced their "regret and concern" at President Trump's decision.
Former US president Barack Obama - whose administration inked the deal - made a rare public criticism of his successor.
"The consistent flouting of agreements that our country is a party to risks eroding America's credibility, and puts us at odds with the world's major powers," he said.