The United States' outsize influence over the global financial system means that its sanctions are powerful tools. The Treasury Department can lock foreign businesses out of the U.S. market if they refuse to comply with sanctions on Iran.
In addition to the United States, Iran negotiated the nuclear deal with China, France, Germany, Russia and the United Kingdom.
France, Germany and the United Kingdom expressed "regret and concern" over Trump's decision and said they intend to preserve the 2015 agreement. In a joint statement on Tuesday, the leaders of the three nations called on Iran to continue implementing the deal, and said Trump should avoid taking any action that would prevent them from implementing the agreement.
"After engaging with the US Administration in a thorough manner over the past months, we call on the US to do everything possible to preserve the gains for nuclear non-proliferation brought about by the JCPoA, by allowing for a continued enforcement of its main elements," the leaders said, referring to the deal by its official name, the Joint Comprehensive Plan of Action.
Trump's plan to restore sanctions and Europe's resolve to maintain the nuclear deal are directly at odds. The success of the sanctions depend on exerting pressure on Iran's economy, while the fate of the deal relies on open access to the Iranian market.
Donald Tusk, president of the European Council, on Tuesday said Trump's polices will be met with a "European approach" that will be discussed at a summit next week.
Iranian President Hassan Rouhani said Iran intends to remain a party to the nuclear deal even as Washington exits, according to a Reuters translation of his speech following Trump's address.
Trump has long threatened to scrap the accord, the signature foreign policy achievement of the Obama administration. However, he certified that Iran was complying with the agreement throughout his first year in the Oval Office, reportedly under pressure from the moderate wing of his administration.
Still, the president began the process of unraveling the deal in October, when he told Congress the agreement was no longer in the country's national security interest. At that point his administration began to push Congress and European allies to reopen negotiations and revise the accord.
When Trump certified Iran's compliance in January, he warned that it would be the last time unless Washington reached a deal with Europe to toughen the terms of the nuclear agreement. That deal has not materialized ahead of the next deadline on May 12.
"Today's action sends a critical message," Trump said. "The United States no longer makes empty threats. When I make promises, I keep them."
"As we exit the Iran deal, we will be working with our allies to find a real, comprehensive, and lasting solution to the Iranian threat," Trump added.
Those efforts will target Iran's ballistic missile program and its role in conflicts throughout the Middle East, according to Trump. Along with those issues, Trump wanted Europe to agree to a deal that would make permanent certain restrictions on Iran's nuclear program that expire in 10-15 years. He also pushed for more intrusive inspections.
"France, Germany, and the UK regret the U.S. decision to leave the JCPOA. The nuclear non-proliferation regime is at stake," French President Emmanuel Macron said in a tweet.
He added, "We will work collectively on a broader framework, covering nuclear activity, the post-2025 period, ballistic activity, and stability in the Middle-East, notably Syria, Yemen, and Iraq."
— CNBC's Jacob Pramuk contributed reporting to this story.