The tweet appeared to refer to comments by Mr. Rouhani, reported by the semiofficial state news agency, warning the Trump administration against continuing hard-line policies against Iran.

“America should know that peace with Iran is the mother of all peace, and war with Iran is the mother of all wars,” Mr. Rouhani was reported saying.

Mr. Trump withdrew from the 2015 Iran nuclear accord in May, and the administration later issued 12 demands for a new deal. U.S. Secretary of State Mike Pompeo warned at the time that the U.S. would impose the “strongest sanctions in history” unless Iran agreed.

Tehran swiftly rejected the terms, which include one requiring a wholesale change in its military posture in the Middle East, where it is backing groups of fighters in Iraq and Syria.

Mr. Pompeo drew parallels to the administration’s “maximum pressure” policy on North Korea, which has been subjected to sanctions intended to force it to give up its nuclear arsenal and ballistic missiles. In Iran’s case, U.S. measures would include pressure on countries to reduce their purchases of Iranian oil to zero, with very few exemptions, made case by case.

Mr. Trump’s Sunday tweet was similar to a threat he made to North Korea in front of reporters last August—of “fire and fury like the world has never seen.” A month later, North Korean leader Kim Jong Un responded, “I will surely and definitely tame the mentally deranged U.S. dotard with fire.”

The 2015 Iran nuclear accord negotiated between Iran, the U.S. and five other world powers was a hallmark of the Obama administration. It codified a trade-off in which Iran agreed to limit its uranium enrichment in return for sanctions relief.

For months, the Trump administration was involved in talks with Britain, France and Germany about strengthening the deal, but the discussions faltered along the way.

European countries earlier this month told Iran they are exploring activating accounts for the Iranian central bank with their national central banks in a bid to open a financial channel to keep the agreement alive.

Iran’s economy is under severe strain, and hundreds of demonstrations have erupted across the country over rising prices, corruption and environmental damage. The value of the country’s currency is down by nearly half since January. Iranians are also increasingly frustrated with the country’s lack of political and social freedoms.

The economic crisis has only grown more severe since the Trump administration pulled out of the 2015 deal: In the month following the May announcement, oil exports were down 16%.

Iran’s foreign minister, Javad Zarif, has called on European countries to introduce measures to ease the impact of the U.S. withdrawal from the deal before U.S. sanctions take effect. The first sanctions are set to be reimposed on Aug. 6, followed by measures targeting Iranian oil on Nov. 4.

Mr. Rouhani, who has sought to make Iran less isolated by forging diplomatic ties with the West, is now seeking to preserve his political future by reaching out to hard-liners who previously opposed his leadership.

Earlier this month, he threatened to disrupt the flow of Middle Eastern oil through the Persian Gulf, a transit route for about a third of the world’s seaborne oil trade. The threat was immediately praised by the country’s military leaders, who have been at odds with Mr. Rouhani over his efforts to improve relations with the West.

On Sunday, Mr. Rouhani again noted Iran’s strategic position on the major oil-trading route, although he stopped short of a fresh threat to cut off shipping, according to the remarks reported by the state news agency.

Mr. Trump’s threat to Iran followed a turbulent week for the president. He faced bipartisan criticism over his handling of a meeting with Russian President Vladimir Putin, and rattled currency and bond markets by attacking the Federal Reserve for signaling plans to gradually raise interest rates.

Write to Jessica Donati at Jessica.Donati@wsj.com