Trump announced the rescue in the early hours of Sunday in a social media post that described the operation in a mountainous area as "one of the most daring" such missions in US history.
The airman, the weapons officer of an F-15 jet shot down on Friday, was wounded but "will be just fine", Trump said on Truth Social.
The jet's pilot was rescued later that day.
Earlier on Sunday Trump ratcheted up pressure on Iran, threatening in an expletive-laden Easter social media post to target Iran's power plants and bridges if the strategic Strait of Hormuz is not reopened.
Trump said his deadline for Iran to open the strait or face attacks on critical infrastructure is Tuesday evening, according to an interview he gave to the Wall Street Journal on Sunday.
The Strait of Hormuz, a conduit for around one-fifth of the world's oil and natural gas supply, has been largely shut down since the war began five weeks ago.
"Tuesday will be Power Plant Day, and Bridge Day, all wrapped up in one, in Iran," Trump said on his Truth Social platform.
"There will be nothing like it!!! Open the F****n' Strait, you crazy bastards, or you'll be living in Hell - JUST WATCH! Praise be to Allah. President DONALD J. TRUMP"
Later on Sunday, the US president in a follow-up post, gave a more precise deadline: "Tuesday, 8:00 P.M. Eastern Time!" (sic)
Adding to the pressure, Washington's ally in the war, Israel, which attacked a major petrochemicals facility on Saturday, was preparing to attack energy facilities next week and was awaiting US approval, a senior Israeli defence official said.
However, in the sort of mixed messaging that has baffled supporters and foes alike, let alone financial markets, Trump on Sunday told Fox News that Iran was negotiating, with a deal possible by Monday.
Tehran is demanding an end to hostilities and its parliamentary speaker, Mohammad Baqer Qalibaf condemned Trump's threats, saying he was being misled by Israeli Prime Minister Benjamin Netanyahu.
"Your reckless moves are dragging the United States into a living HELL for every single family, and our whole region is going to burn because you insist on following Netanyahu's commands," he posted on X.
Thirteen US military personnel have died and hundreds of others have been wounded throughout the Middle East since the US and Israel started operations.
Trump said he will hold a news conference on Monday in the Oval Office alongside the military.
The average US petrol price hit $US4.11 ($A5.95) a gallon on Sunday, according to AAA, up from below $US3 ($A4.30) when strikes on Iran began.
The war, which opened with US and Israeli air strikes across Iran on February 28, has spread into Lebanon, where Israel has resumed its campaign against the Iranian-backed Hezbollah.
Thousands of people have died, mainly in Iran and Lebanon, where Israeli air strikes killed another 11 people on Sunday, according to Lebanon's health ministry.
Trump's blending of religious references with threats of military action drew some criticism on Sunday.
Republican former Representative Marjorie Taylor Greene, writing on X, accused Trump of betraying Christian values. She said Christians in the administration should be "pursuing peace" rather than "escalating war" and argued that Jesus' teachings emphasised forgiveness and love, including toward enemies.
The Council on American-Islamic Relations also condemned Trump's language, saying in a statement that his "mocking of Islam and his threats to attack civilian infrastructure" were reckless and dangerous.
