A US Navy surface drone found and rescued the two crew from the waters of the strait, the US military's Central Command told Reuters.
The pair were rescued within about two hours and were in stable condition, Centcom said in a statement on Tuesday.
It was not immediately clear whether the Apache had been shot down by Iranian fire, experienced mechanical failure, or encountered another problem.
Asked if he knew what had brought the helicopter down, Trump said a report would be issued later on Tuesday.
"The pilots are fine," Trump said, speaking on the runway at New York's John F Kennedy International Airport before returning to Washington, DC.
"Nobody injured."
Iran and Israel said on Monday that they had halted attacks on each other after an appeal from Trump, settling back into a tenuous ceasefire announced on April 8.
Two Iranian air defence personnel were killed in Israeli strikes on Monday, Iran's military said, and their funerals were due to be held on Tuesday afternoon in Tehran.
No deaths were reported in Israel after the Iranian strikes.
Oil prices, which had risen on the exchanges of fire, gave up most of their gains on Tuesday after the two sides paused attacks.
Tehran warned, however, that it would resume hostilities if Israel continued to hit Iran's Hezbollah allies in Lebanon.
On Tuesday, the Israeli military issued an evacuation order for the Lebanese city of Tyre before possible strikes.
Israel's campaign against Hezbollah in Lebanon helped trigger the latest missile exchanges between Iran and Israel - the most direct confrontation since the April ceasefire - complicating Trump's push to end a war that the US and Israel launched on February 28.
Trump also told reporters he might have "an idea" for an Iran deal within a few days, without elaborating.
The Republican president, struggling with record low approval ratings as November's midterm elections approach, has often hinted at an imminent deal with Tehran, but none has yet materialised.
Iran had fired missiles towards Israeli territory late on Sunday, calling the strikes retaliation for attacks on the Iranian-backed Hezbollah militia on the outskirts of Beirut.
Israel then hit Iranian air defence systems and a petrochemical plant that it said was used to produce ballistic missiles.
Iran's Islamic Revolutionary Guard Corps said it retaliated with a strike aimed at a similar Israeli plant in the city of Haifa.
US and Israeli officials said Trump and Israeli Prime Minister Benjamin Netanyahu spoke on Monday.
In an interview with Axios, Trump said he had warned the Israeli leader not to return to war with Iran: "I said, 'Bibi, you better be careful, or you will be on your own very soon.'"
Tehran has long said any peace deal with the US depends in part on an end to fighting in Lebanon, which Israel invaded in March in pursuit of Hezbollah fighters who had fired across the border.
Israel has never halted its Lebanon campaign, which has killed thousands of people, saying the conflict should be treated separately from any US-Iranian ceasefire. Hezbollah has also continued its attacks.
At the same time, Tehran has continued to block most shipping through the Strait of Hormuz, which before the war carried a fifth of the world's crude oil and liquefied natural gas.
Washington has imposed its own blockade of Iranian ports.
Trump has said any peace deal must ensure Iran cannot develop a nuclear weapon.
Iran's demands include the lifting of international sanctions, the release of billions of dollars in frozen assets and recognition of its control of the strait.
