The White House said American Warren Weinstein, a Jewish-American held by al-Qaeda since 2011, and Giovanni Lo Porto, an Italian national held by al-Qaeda since 2012, were killed in a January operation in the border region of Afghanistan and Pakistan.
The operation targeted an al-Qaeda-associated compound and there was no reason to believe either hostage was present at the location, the White House said.
"As a husband and as a father, I cannot begin to imagine the anguish that the Weinstein and Lo Porto families are enduring today," Obama told reporters, with a deep sigh, saying he took responsibility for the deaths and has ordered a full review.
"I profoundly regret what happened," Obama said, explaining he declassified some of the details of the operation so that the families could know what happened.
Earlier, in a written statement, the White House expressed "tremendous sorrow" over the hostage deaths and said the United States had no reason to believe hostages were at the al Qaeda related compound targeted in the operation.
"Analysis of all available information has led the intelligence community to judge with high confidence that the operation accidentally killed both hostages," the statement said.
"No words can fully express our regret over this terrible tragedy," the statement added.
The US believes that Ahmed Farouq, an American who the White House says was an al-Qaeda leader, was killed in the same operation. US officials have also concluded that Adam Gadahn, an American who had served as a spokesman for the terror network, was killed in a separate American operation in January.
The White House says Farouq and Gadhan were not specifically targeted in the operations, nor did the US have information indicating their presence at the sites.
President Barack Obama was set to appear in the White House briefing room to make a statement on the incidents.
Reuters and the Associated Press contributed to this report.