Iranian Foreign Minister Javad Zarif has condemned the US for promising to impose the "strongest sanctions in history" on his country.
Measures outlined by Secretary of State Mike Pompeo, he said, showed the US was a prisoner of its "failed policies" and it would suffer the consequences.
EU foreign policy chief Federica Mogherini also criticised the US.
She said Mr Pompeo had failed to show how dropping the 2015 nuclear deal would make the Middle East safer.
There was, she said, "no alternative" to the agreement, which US President Donald Trump vowed earlier this month to abandon, and she said the EU would stick by it if Iran met its commitments.
Despite the EU's official position, some of Europe's biggest firms who rushed to do business with Iran after the nuclear deal now find themselves forced to choose between investing there or trading with the US.
What did Pompeo announce?
US sanctions lifted after the 2015 deal would be re-imposed, he said, and those and new measures would together constitute "unprecedented financial pressure on the Iranian regime".
He set out conditions for any new deal with Iran, including the withdrawal of its forces from Syria and an end to its support for rebels in Yemen.
The older US sanctions prohibited almost all trade with Iran.
Mr Pompeo did not say what new measures Washington was contemplating but described sanctions imposed last week on the head of Iran's central bank as "just the beginning".
Iran is one of the world's largest oil producers, and the export of oil and gas is worth billions of dollars each year.
Both the country's oil output and its GDP fell noticeably under international sanctions.
The sanctions will not be re-imposed on Tehran immediately but are subject to three-month and six-month wind-down periods.
How else did Iran respond?
Mr Zarif said America was "regressing to old habits" but his country was working with other partners to the nuclear deal in order to find a solution.
Iranian President Hassan Rouhani launched a personal attack on Mike Pompeo, questioning his credibility as a former CIA chief to make decisions for Iran and the world.
A diplomatic smokescreen?
Analysis by Jonathan Marcus, BBC diplomatic correspondent
This is the US "Plan B" for Iran - to step up the sanctions pressure, to force the Tehran government into a new diplomatic deal. It would need to accept broader constraints not just on its nuclear activities but also on its missile programme and wider behaviour in the region.
It is certainly tough but may be totally unrealistic. For sanctions to work they must be comprehensive. The pressure that brought about the JCPOA deal that President Trump has now abandoned was long-standing and widely supported. Now Washington's European allies want to stick with the existing deal. Russia, China and India are unlikely to bow to US pressure.
Compelling allies and other countries to abandon trade with Iran risks damaging a whole series of wider diplomatic relationships.
Critics may charge that this "policy" is impossible - a diplomatic smokescreen intended to cloak a policy whose fundamental goal is regime change in Iran.
Read Jonathan's analysis in full
Why is Iran seen as a regional threat?
It has spread its influence across parts of the Middle East where there are large communities of fellow Shia Muslims, from Iraq to Lebanon.
Its support for Lebanon's Hezbollah movement is particularly alarming for Israel while Saudi Arabia, another bitter enemy, accuses the Iranians of equipping rebels in Yemen.
In the Syrian civil war, it is one of President Bashar al-Assad's few outside allies, sending thousands of fighters and military advisers.
Is the US on its own?
Israel praised the Trump administration's decision to pull out of the pact but the move was roundly criticised by fellow signatories, including France, Germany, the UK and Russia.
All of the above signatories pledged to honour their commitments under the deal.
Mr Pompeo has made clear he expects the backing of his allies in Europe but also called for support from "Australia, Bahrain, Egypt, India, Japan, Jordan, Kuwait, Oman, Qatar, Saudi Arabia, South Korea [and] the UAE".
What was agreed under the 2015 deal?
The Joint Comprehensive Plan of Action (JCPOA) saw Iran agree to limit the size of its stockpile of enriched uranium - which is used to make reactor fuel but also nuclear weapons - for 15 years and the number of centrifuges installed to enrich uranium for 10 years.
Iran also agreed to modify a heavy water facility so it could not produce plutonium suitable for a bomb.
In return, sanctions imposed by the UN, US and EU that had crippled Iran's economy were lifted.
The deal was agreed between Iran and the five permanent members of the UN Security Council - the US, UK, France, China and Russia - plus Germany.
Iran insists its nuclear programme is entirely peaceful, and its compliance with the deal has been verified by the IAEA.
