Two weeks after the collapse of an interim truce meant to end the war, the Iranian armed forces responded by firing at US bases in neighbouring Arab countries and warning people there that they might strike non-military buildings used by US personnel.
The Iranian army said Trump's threats only strengthened its resolve.
The Houthis, who control northern and western Yemen, said this week they were imposing a naval blockade on Saudi Arabia, which has diverted millions of barrels of oil each day by pipeline to the Red Sea to skirt Iran's near-total blockade in the Strait of Hormuz.
The strait is the focus of a war that has killed thousands in almost five months, stoked global inflation and fanned fears of an economic downturn.
The Houthi fighters struck two Saudi tankers on Thursday in the Red Sea's Bab el-Mandeb strait, pushing oil prices over the $US100-a-barrel level for the first time since May, although they eased to under $US98 on Friday.
The attacks forced several other tankers to turn around and head north through the Suez Canal, potentially using a much longer and more expensive route to reach Asian customers by sailing around Africa.
Trump wrote on social media he would hold Iran accountable for any further attacks by the fighters, "and major military punishment will be inflicted upon Iran and, of course, the Houthis, themselves".
He told Axios he was considering relaunching major combat operations in Iran and was close to a decision.
"They haven't received enough pain yet," the US news outlet quoted Trump as saying.
Iranian state media said missiles had struck Qeshm Island on the Strait of Hormuz.
The Iranian army said it had attacked US military equipment depots at Camp Buehring in northern Kuwait, and the positions of US troops at Camp Arifjan and at Camp Doha, near Kuwait City.
Iran's Revolutionary Guards said they had significantly damaged a surveillance tower used by the US Fifth Fleet in Bahrain.
US air strikes hit Piranshahr, in West Azerbaijan Province near Iran's border with Iraq, on Friday, damaging several vehicles, according to the province's emergency services.
US missiles struck a Revolutionary Guards Navy headquarters in the Ziba Kenar area of the Caspian Sea province of Gilan, according to a security official in Gilan quoted by Iranian media.
Gilan provincial authorities said US projectiles had also struck the port city of Bandar Anzali.
Four people were killed and five injured in a US missile attack on the Iranian city of Ahvaz, state broadcaster IRIB said.
In a statement, the Revolutionary Guards warned the public in the Gulf countries of potential strikes against US military personnel who might be using "buildings in cities as locations to direct their crimes".
Since the US and Israel began the war in February, Iran has demonstrated that it can hit US targets in the region, even though Trump said early on that the US had wiped out Iran's military capabilities.
High oil prices are fuelling inflation globally, putting Trump's Republican allies under pressure ahead of midterm congressional elections in November.
The Republican-led US House of Representatives backed legislation on Thursday directing Trump to halt US military action against Iran, its latest rebuke of the president.
Congress alone has the power to declare war under the US Constitution.
Repeated similar resolutions have not led to cessation of the war.
The Senate voted hours later to block a separate, similar measure.
