The Connecticut event was closed to the news media but the remarks were caught on camera and posted online, and Republican officials confirmed Steele had made the comments.
Steele further said of Obama: "If he's such a student of history, has he not understood that, you know, that's the one thing you don't do is engage in a land war in Afghanistan? All right?
"Because everyone who's tried, over a thousand years of history, has failed. And there are reasons for that. There are other ways to engage in Afghanistan."
As criticism swelled on Friday, Steele issued a statement stressing his support for US troops.
"There is no question that America must win the war on terror... And, for the sake of the security of the free world, our country must give our troops the support necessary to win this war," he said.
"The stakes are too high for us to accept anything but success in Afghanistan."
But he did not acknowledge his factual error about a war launched by George Bush, the Republican US president who preceded Obama, in response to the attacks of September 11, 2001 on the US.
The US and its allies overthrew Afghanistan's Taliban government shortly after the 9/11 attacks on New York and Washington DC but the war lagged as the US shifted focus to Iraq.
Obama shifted the focus back to Afghanistan when he took office early last year and has committed tens of thousands more troops.
Steele's comments came as General David Petraeus, arrived in Afghanistan on Friday to take over the war after Obama dismissed General Stanley McChrystal as commander of US and Nato forces there last week over disparaging remarks he and his aides made about administration officials in an interview.
Steele called the dismissal "very comical" but said it showed the frustration members of the military had with Obama.
Calls for resignation
Conservative columnist Bill Kristol, writing for The Weekly Standard, was among the first to say Steele should resign.
"There are, of course, those who think we should pull out of Afghanistan, and they're certainly entitled to make their case," wrote Kristol, a consistent supporter of the Afghanistan war.
"But one of them shouldn't be the chairman of the Republican party."
Erick Erickson, the editor of the popular conservative website RedState.com and an opinion leader among younger Republicans, also called for Steele to go.
"Michael Steele must resign. He has lost all moral authority to lead," he said.
Doug Heye, a spokesman for the Republican National Committee, said Steele "clearly supports our troops but believes that success of the war effort in Afghanistan requires the ongoing support of the American people".
"The responsibility for building and maintaining that strategy falls squarely on the shoulders of the president," he added.
Members of Obama's Democratic party, wasted no time in pouncing on Steele's gaffe as they geared up for midterm elections in November, when voters will pick 37 governors, 36 senators and the entire 435-member US House of Representatives.
Brad Woodhouse, a spokesman for the Democratic National Committee, said it was "simply unconscionable that Michael Steele would undermine the morale of our troops when what they need is our support and encouragement".
"Michael Steele would do well to remember that we are not in Afghanistan by our own choosing, that we were attacked and that his words have consequences."
Steele has been prone to gaffes that have enraged congressional Republicans.
In the last year, he predicted that Republicans would not win control of the House of Representatives in the November vote.
He also drew Republican ire when he criticised fellow Republicans in a book and then told critics of the book to "get a life" and "shut up".
Earlier this year, his oversight of the Republican National Committee was called into question because of lavish spending, including money to entertain donors at a lesbian bondage club in Los Angeles.
That incident led to the departure of a key Steele adviser, the party's finance chief and a senior committee staff member.