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)
In the kind of mixed messaging that has baffled supporters, foes and financial markets alike, Trump told Fox News on Sunday that Iran was negotiating, with a deal possible by Monday.
Later on Sunday, Axios reported the United States, Iran and regional mediators were discussing terms of a potential 45-day ceasefire that could lead to a permanent end to the war, citing four US, Israeli and regional sources with knowledge of the talks.
Reuters could not immediately verify the report. The White House and the State Department did not immediately respond to Reuters' requests for comment.
The mediators were negotiating a two-phased deal, the report said. The first phase would be a 45-day ceasefire; the second would be an agreement on ending the war.
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.
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.
