The Iranian navy said it had fired after “several ships attempted to travel along an unapproved route” and ignored directions to transit through Iran’s territorial waters. It warned that it would meet any U.S. retaliation to its attack with a “forceful response.”

Iran’s latest move came after a heavy week of skirmishes between the United States and Iran focused on the strait. Tehran’s military has continued to claim its own waters as the only viable route, and this week, U.S. officials said, Iran attacked three ships traveling through the waterway. Iran did not claim responsibility for those attacks.

Those earlier strikes prompted a forceful American retaliation and U.S. officials to issue the ultimatum, saying that they expected Iran to publicly declare an end to attacks on commercial ships in the waterway and acknowledge that all channels for crossing it were open. If not, the officials said, “we’re not going to have a good outcome for them.”

Adding to the tensions, President Trump and Iran’s supreme leader had continued to trade threats. In a rare written message on Saturday, Ayatollah Mojtaba Khamenei vowed revenge for the killing of his father and predecessor, Ayatollah Ali Khamenei, calling it the “demand of our nation.”

It was his first statement since a weeklong funeral for the elder Ayatollah Khamenei, culminating in his burial in Mashhad on Thursday. Although many Iranians loathed his repressive 37-year rule, millions joined processions across Iran and Iraq, which the Islamic republic used to project unity and defiance toward its enemies.

The ayatollah’s words deepened the doubt hanging over the fragile truce. While the Trump administration had said the cease-fire would fully lift Iran’s blockade of the Strait of Hormuz — which began after the U.S.-Israeli war on Iran in February — that has yet to happen.

Instead, Iranian forces have fired on ships traveling on routes they deem unacceptable, seeking to maintain control of the strait, and prompting the U.S. military to retaliate with attacks on Iranian military sites. The United States also revoked a sanctions waiver that had temporarily permitted the sale of Iranian oil.

So far, threats by President Trump and bombardment by American warplanes have not succeeded in forcing Iran to relax its grip on the strait. Instead, both countries are now engaged in a simmering standoff — neither all-out war nor peace — with Mr. Trump saying on Friday that June’s cease-fire agreement was “over.”

Later, Mr. Trump added a threat on social media, vowing that “1000 Missiles are Locked and Loaded and aimed at the Islamic Republic of Iran” should the country make good on its threats to assassinate him or try to do so. U.S. law enforcement says Iran has devised several plots to kill him, as well as other American government officials, over the years.

In a seeming retort, Iran’s supreme leader, Ayatollah Khamenei, said that the country would seek revenge for his father. “We pledge that we will avenge your pure blood and the blood of all those martyred,” he wrote in a message published on Saturday.

The new supreme leader was absent from his father’s funeral and has been neither seen nor heard in public since succeeding him in early March. He is believed to have been wounded on the first day of the war, according to Iranian and Israeli officials.

The latest round of fighting began this week after Iran fired on three ships traversing the strait, U.S. officials said, with some of those vessels linked to the Persian Gulf countries of Qatar and Saudi Arabia, both U.S. allies.

U.S. forces retaliated with two days of heavy bombardment, attacking about 170 Iranian military targets, according to the U.S. military. On Friday, an Iranian health ministry spokesman said on social media that at least 17 people were killed in the attacks. Iran’s state television identified at least eight of the dead as soldiers.

Iran fired waves of ballistic missiles and drones at Kuwait, Bahrain and, for the first time since the beginning of the truce, Jordan.

The battle of wills in the Persian Gulf has reinforced skepticism that Mr. Trump can reach a broader pact with Iran to rein in its nuclear program, a core aim of the U.S.-Israeli war. Under the June cease-fire agreement, the two countries were supposed to negotiate a more comprehensive agreement in 60 days — a time frame that appears increasingly remote.

Oman has circulated its own proposal for jointly administering the Strait of Hormuz alongside Iran, including the potential imposition of service fees on transiting ships. The plans would be a significant change from the prewar status in the strait, when boats generally passed freely.

Leily Nikounazar and David E. Sanger contributed reporting.