"During his tenure at Centcom, Fallon's job has been to help ensure that America's military forces are ready to meet the threats of an often troubled region of the world, and he deserves considerable credit for progress that has been made there, especially in Iraq and Afghanistan," Bush wrote.
Fallon is in Baghdad, making the rounds to speak to all of his commanders.
ABC News' Martha Raddatz spoke to Fallon Tuesday as he waited to go in and see Gen. David Petraeus, the U.S. commander in Iraq.
Fallon told ABC News that he sensed the Esquire article angered members of the Bush administration.
A senior administration official met with Gates as soon as the article came out and was very worried about the reaction from the White House at that time.
Fallon told Raddatz he is grateful for the way Gates handled his resignation.
Democrats on Capitol Hill seized on the resignation as more evidence that the Bush administration "silences opposing voices."
"Yet another example that independence and the frank, open airing of experts' views are not welcomed in this administration," read a statement from Senate Majority Leader Harry Reid.
"It is also a sign that the administration is blind to the growing costs and consequences of the Iraq War, which has so damaged America's security interests in the Middle East and beyond," Reid said.
Fallon, who has had a 41-year career in the Navy, took the central command post March 16, 2007, succeeding Army Gen. John Abizaid, who retired. Fallon previously served as commander of U.S. Pacific Command.
His resignation was applauded by Adm. Michael Mullen, chairman of the Joint Chiefs of Staff.
"I support Secretary Gates' decision to accept Adm. Fallon's letter of resignation and request for retirement. I also respect the reasons for which Adm. Fallon submitted it and applaud his ability to recognize the responsibility before him," Mullen said in a written statement.
"He had an enormous impact not only on the way we operate and fight in this new century, but also on the way in which we stay engaged globally," Mullen's statement read.
Gates said that until a permanent replacement is nominated and confirmed by the Senate, Fallon's place will be taken by his top deputy, Army Lt. Gen. Martin Dempsey.
ABC News' Zach Wolf, Ann Compton, John Cochran and Jennifer Duck contributed reporting.