WASHINGTON, Oct. 18 — The House today upheld President Bush’s veto of a bill to provide health insurance to 10 million children, but Democrats vowed to send it back to him next month, with minor changes, in the belief that they could ultimately prevail.

Despite a multimillion-dollar advertising campaign and intense lobbying by children’s advocates, supporters of the bill were unable to convert a single House Republican who voted against passage of the bill last month.

For now, the insurance vote stands as the latest example of how Mr. Bush can still get his way on Capitol Hill. Through artful use of veto threats and his veto pen, Mr. Bush has fended off attempts to force a change of course in Iraq — a feat Democrats would never have imagined when they pushed Republicans out of power a year ago. He has twisted Democrats into knots over domestic surveillance, and forced them to rethink a resolution condemning as genocide a century-old massacre of Armenians.

The outcome today, reminding Democrats of the limits of their power, came as Congress and the president prepared to square off over a dozen spending bills needed to finance the government in the new fiscal year. President Bush has threatened to veto at least 10 of those measures, while also holding the Democrats responsible for not acting more quickly on the bills, which were supposed to be enacted by Sept. 30.