Returning to a central issue of the 2018 campaign, House Democrats on Thursday passed legislation to reverse Trump administration rules that allow expansion of health care plans that do not have to comply with the Affordable Care Act’s mandated coverage of pre-existing medical conditions.

The vote — 230 to 183 — was a jab at President Trump, who has pressed for ways around the coverage mandates of President Barack Obama’s signature domestic achievement while claiming he is committed to protecting Americans with chronic illnesses.

But it served a larger political purpose, kicking off a push by House Democrats on health care, an issue they see as central to winning back the White House and holding their gains in the House in 2020. Over the next two weeks, Democrats expect to pass a raft of legislation to drum home the point that, even as they clash with the White House over the findings of the Mueller report, they will continue to focus on proposals that help real people.

The House bill would roll back an October 2018 regulation by the Treasury Department and the Department of Health and Human Services that expanded the eligibility of states to receive waivers to the Affordable Care Act to “increase coverage options” — particularly with insurance plans that offer more limited coverage than those compliant with coverage mandates under the 2010 health law.

Another Democratic bill, due for a vote next week, would eliminate a Trump administration rule change that expanded the time limit for so-called limited-duration insurance plans to as long as three years from three months — plans Democrats call “junk insurance.”

In all, a dozen health care bills are scheduled for votes over the next two weeks, a blitz intended to establish the House Democrats’ record on health care heading into next year’s election — and possibly divide congressional Republicans, who have maintained that they too want to preserve protections for people with pre-existing health conditions or chronic ailments.

Other measures include a restoration of funding used to help consumers find and sign up for coverage under the health law, new disclosure requirements for prescription drug rebate programs, several measures intended to reduce drug costs and increase access to advanced medications known as biologics, and the allocation of $10 billion per year in new “reinsurance” payments that would help lower health care premiums by offsetting the costs of the most expensive insurance claims.

Taken in total, the package is intended to build on the system established by the health law without disrupting the health care system for Americans who get coverage through work or a government program.

It also represents a core component of Speaker Nancy Pelosi’s policy agenda at a time when she has rebuffed calls to embrace more sweeping proposals, including the single-payer “Medicare for all” plan supported by many party liberals, including Representative Alexandria Ocasio-Cortez of New York.

Ms. Pelosi is adopting a more measured approach. Some bills, like the pre-existing conditions package, are aimed at forcing Republicans into taking tough votes on health care. Others, like the drug bills, are expected to draw some Republican support — and could even win the support of Mr. Trump, who has said he wants to tackle the cost of prescription medications.

“Every speech by Republicans in 2018 stated that they wanted to protect patients with pre-existing conditions, even as they were maligning the A.C.A.,” said Representative Ann McLane Kuster, Democrat of New Hampshire, who introduced Thursday’s bill.

“I think, symbolically, this bill is the one that frames the rest of the issues, and our focus on expanding health care,” she said. “It’s also about pushing back on this administration’s attempts to sabotage the Affordable Care Act at every step along the way.”

The short-term plans targeted by Democrats were initially intended as stopgaps for people who had gaps their coverage, not as long-term sources of coverage. In the past two years, they have become one of the administration’s main avenues of attack on the health law.

Mr. Trump has said he is merely trying to lower insurance costs for people who want to take the risk of not having the comprehensive coverage mandated by the health law.

As of yet, however, no states have applied for the waivers targeted by Ms. Kuster’s bill, and several states have reinstated the three-month limits the Trump administration hoped to extend.

Still, the rule changes are seen as a major threat to the survival of the health law; as many as 1.6 million health care consumers could opt out of the law’s regulated marketplace by 2022, according to officials with the federal Centers for Medicare and Medicaid Services.

Rolling back these rules — and restoring the existing system under the health law — would save taxpayers an estimated $8.9 billion over the next decade, realized by keeping relatively healthy people, who draw fewer benefits in the coverage pool, from leaving the federally subsidized system, according to the nonpartisan Congressional Budget Office.

Ms. Kuster’s bill, which four Republicans crossed party lines to support, has virtually no chance of passing in the Senate. The White House budget office defended the granting of state waivers this week, arguing that the rule changes have not resulted in the loss of coverage for people with pre-existing conditions.

“The president has repeatedly made clear that this administration will protect people with pre-existing conditions,” a statement from the office said. “Coverage in the individual market has become increasingly unaffordable for unsubsidized consumers, including those with pre-existing conditions. The 2018 guidance freed states to develop health coverage that best fits the needs of their residents.”

Supporters of the bill say that the rule changes are a thinly veiled attempt to weaken the health law’s online insurance exchanges by enticing younger, healthier patients to withdraw — making them financially unsustainable.

“The people who enroll in short-term plans often wind up not getting the insurance they were counting on, even though they seem superficially attractive for younger people because the premiums are lower than Obamacare,” said Stan Dorn, a health policy analyst with Families USA, a liberal health care advocacy group that has strongly supported the health law.

“It’s also devastating to the system to pull these young and healthy people out, because the premiums rise for everybody else,” he added.

Mr. Trump, frustrated in his attempts to repeal the health law in Congress, has tried to attack it piecemeal, by loosening regulations that govern insurance plans and by granting waivers to states that weaken the complex system of exchanges and subsidies needed to make it function.

This month, the administration formally declared its opposition to the entire Affordable Care Act, arguing in a federal appeals court filing in Texas that the system — which covers about 23 million low-income and working-class Americans — should be scrapped.

Democrats are hoping to cast doubt on Mr. Trump’s vow to “always protect Americans with pre-existing conditions” — but they are also hoping the push will mute the differences in their own party. On Wednesday, the second-ranking Democrat in the House, Representative Steny H. Hoyer of Maryland, played down divisions between Ms. Pelosi and liberals who are pushing for a vote on the Medicate for All Act.

“Speaker Pelosi and I have both maintained that the best way for us to get there — and we want to get there — is to make sure that the A.C.A. is working as it was intended to work,” he told reporters during his weekly pen-and-pad session.