WASHINGTON — President Obama, beginning his final push for a health care overhaul, called Wednesday for Congress to allow an “up or down vote” on the measure, and sketched out an ambitious — and, some Democrats said, unrealistic — timetable for his party to pass a bill on its own within weeks.

“I believe the United States Congress owes the American people a final vote on health care reform,” Mr. Obama said during a 20-minute speech in the East Room of the White House. He said there was no point in starting over, as Republicans are demanding, and called on nervous Democrats to stick with him, declaring there was no reason “for those of us who were sent here to lead to just walk away.”

The speech, less than a week after Mr. Obama held a high-profile televised health care forum, will usher in what White House officials say will be their last campaign to bring Washington’s long and contentious health care debate to a close — with a bill-signing ceremony at the end.

On Thursday, Health and Human Services Secretary Kathleen Sebelius will meet at the White House with insurance industry executives to spotlight unpopular rate increases; next week, Mr. Obama will travel to Missouri and Pennsylvania to stump for the health care bill.