A first phase of US sanctions against Iran went into effect midnight, targeting Iran's access to USA banknotes and key industries including cars and carpets.
In his statement, Trump said the United States "is fully committed to enforcing all of our sanctions, and we will work closely with nations conducting business with Iran to ensure complete compliance".
John Glaser, director of foreign policy studies at the Cato Institute, noted that the U.S. sees the sanctions "as a tool to pressure Iran to come back to the negotiating table to rehash the nuclear deal on terms more to Trump's liking". European allies said they "deeply regret" the U.S. action.
As of midnight, harsh sanctions on Iran have kicked in as part of the administration's strategy to keep Tehran from achieving its goal of building a nuclear bomb.
"There is a clear interest around the world", one staffer said. But, he said, the reimposed sanctions would endanger "ordinary Iranians". But on Iran, Mr Trump now has a team who share his contempt for the regime and are even more determined to see its end.
Brussels says protecting businesses based there is necessary because the sanctions have an "unlawful" reach beyond U.S. borders.
He said Iran's leaders may need to "drink the poison cup" and negotiate with the US.
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European officials have condemned the renewed sanctions as illegal and in violation of a UN Security Council resolution, vowing to counter the U.S. measures and save the Iran nuclear accord.
"Anyone doing business with Iran will NOT be doing business with the United States", Trump vowed on Twitter.
Mr. Erdan was speaking after European Union foreign policy chief Federica Mogherini said the bloc is actively encouraging enterprises to increase their business with Iran.
Other countries have also asked European Union officials for details on the blocking regulations, as they explore ways to bypass sanctions.
He said: "If you stab someone with a knife and then you say you want talks, then the first thing you have to do is remove the knife".
The US has long designated Iran as the world's foremost state sponsor of terrorism, Pompeo noted Sunday, adding that the Islamic Republic can not expect to be treated as an equal in the global community until it halts such activities. But most Iranians have yet to see major economic improvement, and the prospect that Washington would re-impose sanctions helped drive a collapse in the value of Iran's currency this year, raising the cost of imports.
"If Iran were really serious they'd come to the table".
But many large European firms are leaving Iran for fear of U.S. penalties, and Trump warned of "severe consequences" for firms and individuals that continued to do business with Iran.
"There will be pressure because of sanctions, but we will overcome this with unity".