Washington ? Six months in office, President Barack Obama sought Wednesday night to rally support for sweeping health care legislation he’s struggling to push through Congress, expressing support for a surtax on families making more than $1 million a year to help pay for it.

Under pressure from Democrats to weigh in personally on the details of legislation, Obama also vowed at a prime-time news conference to reject any measure “primarily funded through taxing middle-class families.”

While the session was dominated by health care, Obama said in response to one question that Cambridge, Mass., police “acted stupidly” last week in arresting Henry Louis Gates Jr., a black scholar at Harvard, in his house. Police were called to the house to investigate a possible break-in. Gates produced identification but was arrested on a charge of disorderly conduct after protesting the police conduct.

“Now I don’t know, not having been there, what role race played in that,” the president said. But he added that blacks and Latinos often are stopped by authorities in disproportionate numbers. Police have dropped charges initially lodged against Gates.

Obama stepped to the microphone looking grayer than the man who ran for president and took office in January and immediately began confronting the worst economic recession in decades.

He defended his decision to set a midsummer deadline for the House and Senate to act on health care, even if it isn’t met. “I’m rushed because I get letters every day from families that are being clobbered by health care costs, and they ask me can you help,” he said.

If the consequences are high for nearly 50 million Americans who lack insurance, the political impact is huge for Obama, who is putting much of his credibility on the line to gain congressional passage. His stepped-up public role comes as he faces rising criticism from Republicans, sliding public approval ratings and divisions within his party. Obama acknowledged that many people are uneasy about growing federal budget deficits and the fast-rising government debt.

He said that without a deadline for action, a recent proposal to curtail the growth in Medicare costs would not have materialized “until who knows when.” He said in the past few days, leaders in both houses had agreed to incorporate it into legislation taking shape.

Asked if it was his job to produce a deal on legislation, the president said: “Absolutely it’s my job. I’m the president. And I think this has to get done.”

He said that since he moved into the White House, “we have been able to pull our economy back from the brink.”

Yet, he said, “of course we still have a long way to go.” Obama didn’t say so, but unemployment, currently 9.5 percent, is expected to remain stubbornly high for many months to come.

He was eager to talk about health care — an issue that now towers above all others — and has led at least one Republican to say that it could prove to be the president’s Waterloo if the drive collapses.

“This isn’t about me. I have great health insurance, and so does every member of Congress,” he said.

The president said that in addition to helping millions who lack coverage, the health care legislation is central to the goal of eventually rebuilding the economy stronger than it was before the recession that began more than a year ago.