Mr. Obama’s themes echoed his State of the Union address in January and his speech in Osawatomie, Kan., in December, when he invoked a Republican president, Theodore Roosevelt, who he said combined a fervent belief in the free market with a resolve to protect those vulnerable to its excesses.

But the president turned a harsh new spotlight on the 2013 budget, drafted by Representative Paul D. Ryan, a Wisconsin Republican who chairs the Budget Committee. The proposal, he said, calls for across-the-board cuts in discretionary spending, as well as tax cuts, which he said would disproportionately benefit households earning more than $250,000 and would cost $4.6 trillion over the next decade.

“Disguised as deficit reduction plans, it is really an attempt to impose a radical vision on our country. It is thinly veiled social Darwinism,” Mr. Obama said. “And by gutting the very things we need to grow an economy that’s built to last — education and training, research and development, our infrastructure — it’s a prescription for decline.”

Singling out Medicare, the president asserted that the Republican plan to shift people to a system of vouchers would drive up the cost of health care for the elderly, since private insurance companies would target the youngest and healthiest people and leave the rest to rely on Medicare.

For millionaires, the president said, the average annual benefit of the tax cuts would be $150,000 — money that he said could be used to pay for computer labs in schools, salaries for police officers and firefighters, medical care for veterans and a year’s worth of prescription drugs for older people.

The White House’s calculation for the tax benefit is straightforward, but Republicans on the House Budget Committee say it is wrong. The average household earning more than $1 million would gain $46,000 from the House budget’s repeal of the Medicare hospital insurance tax that was part of the health care law, the Republicans said, and $105,000 from the extension of the Bush-era tax cuts that Mr. Obama wants to see expire next year.