“For the first time, since 2010, we are on offense on health care and we are going to prosecute a campaign on offense to ensure the law never faces the threat of repeal,” said Brad Woodhouse, a veteran Democratic operative who is joining the group to oversee the day-to-day operations on behalf of the health care law.

Mr. Woodhouse said the Trump administration underscored the need for the group’s new mission last week, when officials announced that they plan to slash the government’s advertising budget for encouraging people to enroll in the health care marketplaces created by the Affordable Care Act. Officials at the Department of Health and Human Services said the advertising budget for the open enrollment period that starts in November would be cut to by 90 percent, to just $10 million.

Democrats believe the reduction is an effort to destabilize the already shaky insurance markets by making sure that fewer people sign up for coverage. Officials countered that the spending was not necessary because most Americans already know about that the health insurance options that are available.

President Trump and Republicans have made no secret of their desire to get rid of the health care law. And they have repeatedly said they believe the insurance markets are failing. In March, Mr. Trump tweeted that “Obamacare is imploding. It is a disaster and 2017 will be the worst year yet, by far!”