This is assuming he isn't popular, which is not true. Otherwise, it's a fair analysis. Or not. :D
http://www.tnr.com/blog/jonathan-chait/76314/there-anything-obama-could-have-done-remain-popularIs There Anything Obama Could Have Done To Remain Popular?
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Okay, but what if Obama never promised to reform health care in the first place? Suppose furthermore that he enacted a much smaller stimulus, and let's suppose that conservatives are right that such a plan would have helped the economy as much or more. And suppose that he decided not to pursue financial reform -- or, at least, pursued a plan that gained the assent of Wall Street and, thus, many Republicans. Would he be more popular now?
Let me answer this question with another question: Am I the only person who remembers George H.W. Bush? He presided over a recession, albeit a far milder one. And he was lambasted for lacking an agenda. The notion that Bush lacked a domestic agenda was the defining indictment of his presidency. Here's a very typical piece of analysis, also from the New York Times, from April 30, 1992:
Fifty-eight percent of the 1,151 registered voters questioned April 20-23 said the Administration was drifting without clear policies. More than a third said it was moving carefully to develop its plans, a reversal from April 1989, at the start of the Bush Presidency, when 61 percent said the Administration was moving carefully. ...
Acknowledging that Mr. Bush had not yet given Americans a vision of where he would take the country in a second term, Marlin Fitzwater, the President's spokesman, said, "By the time of the convention we want to have laid out an intellectual framework of what's been accomplished, what needs to be reformed and our vision of the future."
Even if Obama had the political space with Democrats to abandon his campaign platform, and even if he could have enacted some non-controversial stimulus without any economic ill effects, voters would still be upset over the economy. The accusation would simply be that he's an impotent bystander to the crisis. It's certainly true that voters disapprove of the stimulus. But that's no reason to assume that Obama had any more politically expedient course of action at hand. Different world leaders have tried all sorts of domestic approaches during the economic crisis, and pretty much all of them have seen their popularity fall.