WASHINGTON — The House voted Wednesday to repeal the Democrats’ landmark health care overhaul, marking what the new Republican majority in the chamber hailed as the fulfillment of a campaign promise and the start of an all-out effort to dismantle President Obama’s signature domestic policy achievement.

The vote was 245 to 189, with 3 Democrats joining all 242 Republicans in support of the repeal.

Leaders of the Democratic-controlled Senate have said that they will not act on the repeal measure, effectively scuttling it.

While conceding that reality, House Republicans said they would press ahead with their “repeal and replace” strategy. But the next steps will be much more difficult, as they try to forge consensus on alternatives emphasizing “free market solutions” to control health costs and expand coverage.

Even as four House committees begin drafting legislation, Republicans said they would seek other ways to stop the overhaul, by choking off money needed to carry it out and by pursuing legislation to undo specific provisions, like a requirement for most Americans to carry health insurance or face penalties. The law is also under challenge in the federal courts, with the individual coverage requirements fueling a constitutional battle likely to be decided by the Supreme Court.