If confirmed by the US Senate, Dempsey would replace Admiral Mike Mullen as chairman of the Joint Chiefs of Staff when he steps down on October 1.
Obama made the high-profile pick in a White House ceremony just before heading to Arlington cemetery for a Memorial Day event to honour America's war dead.
The selection comes as Obama is facing growing pressure to accelerate a troop drawdown in Afghanistan and is preparing to complete a withdrawal from Iraq.
Dempsey is currently the Army Chief of Staff. Obama said he was naming General Raymond Odierno to succeed Dempsey in that post.
Admiral James Winnefeld, commander of US Northern command, was named as vice chairman of the Joint Chiefs of Staff.
AFP adds
General Martin Dempsey has led soldiers in combat in Iraq and is keenly aware of the growing strain on the force after years of war.
An Irish-American who taught English literature to cadets at West Point, the 59-year-old graduated from the same class at the military academy as another four-star general, David Petraeus.
Until now, Petraeus, who rose to fame as commander in Iraq and will be stepping down soon as chief in Afghanistan to take over the CIA has largely overshadowed his former classmate.
Apart from Petraeus, Dempsey had another prominent classmate as a younger officer.
At the US Army's staff college in the 1980s, Dempsey got to know a Pakistani officer, Ashfaq Kayani, now Pakistan's powerful army chief who has clashed with the United States in the aftermath of the raid that killed Osama bin Laden.
As commander of the 1st Armored Division in Iraq in 2003-2004, Dempsey oversaw tanks and troops that fought insurgents led by Shiite cleric Moqtada al-Sadr, pushing the militia out of southern cities.
In his new job as chairman of the Joint Chiefs of Staff, Dempsey's two combat tours will help shape how he advises the president as the United States wraps up its mission in Iraq and begins to withdraw some of the 100,000 American troops in Afghanistan.
"One of things he'll bring to the job is current experience in the wars that we're in," said David Barno, a retired lieutenant general and senior adviser at the Center for a New American Security.
Dempsey is due to take over as chairman at a time of turmoil in the Arab world and growing pressure on the defense budget, with the Pentagon bracing for cuts and a possible scaled back force.
Unlike the outgoing chairman Admiral Mike Mullen, a reserved figure with a soft-spoken manner, Dempsey is an extrovert with an irreverent sense of humor and a penchant for singing in public.
Proud of his working class roots in New York and New Jersey, Dempsey's favorite tune is Frank Sinatra's "New York, New York," which he belts out with gusto in unabashed performances captured on YouTube.
Dempsey is known as a thoughtful, independent-minded leader who shuns formalities and goes out of his way to hear from junior officers, said Barno, a friend and former colleague.
At Central Command, Dempsey was "known for going to the back rows of the auditorium when he was giving his morning updates, asking junior officers what they thought," Barno said.
He has "an ability to feel the soul of the organization, to really feel what young soldiers and young leaders are going through and to connect with them in ways better than almost anyone I know," Barno told AFP.
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