Media reports said the attack killed one person and wounded four.
Iran's state-run IRNA news agency quoted the Qeshm governor as blaming a "terrorist enemy" for Sunday's attack.
There was no immediate comment by the US military, which has struck Iranian-flagged vessels during its blockade of Iranian ports and faces a new threat from Iranian ballistic missiles launched at its warships.
The United Kingdom Maritime Trade Operations monitor said a vessel was hit by a projectile while transiting the strait although it was not clear if this was referring to the same attack.
The monitor said a fire had broken out and that local authorities were helping those on board leave the vessel.
Qeshm island, about 22km from the port city of Bandar Abbas, is key to Iran asserting control over the Strait of Hormuz.
Hours before the attack, Iran's government remained defiant.
"Our people can't be bullied into submission. Iran won't surrender," President Masoud Pezeshkian said in a social media post.
Iran has said foreign ministers from regional countries would meet on Monday in Oman to discuss efforts by Iran and Oman, located on the other side of the strait, to manage shipping on it.
But the foreign minister of Oman, Badr Albusaidi, said on X late on Sunday that the meeting had been postponed "in the interests of consensus".
Only Iraq had publicly confirmed its attendance while Bahrain had said it would not attend, citing a lack of diplomatic relations with Iran.
There had been some optimism in the region about the meeting, with hope that Gulf countries might use it to assert a collective response to a war they had no part in starting.
For months, they have faced Iranian attacks that Iran insists were targeting US military assets on their lands.
A Gulf source told Reuters before the announcement of a delay that Saudi Foreign Minister Faisal bin Farhan would attend and Qatar and the United Arab Emirates were also expected to participate in the talks aimed at agreeing a temporary mechanism for the strait and maybe also longer-term arrangements.
Commercial ship traffic remains low in the Strait of Hormuz as attacks continue on what was seen as an international waterway before the US and Israel launched the war on February 28.
The US has asserted that plenty of oil is getting out as its military helps to guide ships along Oman's coast, at the risk of being struck.
But Iran has said reaching an agreement with Oman on the strait does not mean it will open for shipping.
"The waterway would be opened up on the condition that the US will terminate its blockade," Iran's president told broadcaster India Today TV on Saturday.
US President Donald Trump reiterated on Sunday that he expected the Iran war to end this year, possibly just after midterm elections due in the United States in November, resulting in a drop in oil prices.
"Iran will end right after - maybe before - but it'll end right after the midterms. Gasoline's going to drop like a rock," Trump told reporters at the Irish Open golf championship.
"I will tell you this: Iran wants to make a deal so bad. They are calling constantly. They want to make a deal. They've got to make the right deal. I'm not going to make a deal that's no good," he said.
with Reuters
