“They will have, if they get this nominee, a Supreme Court ready to rule against protections for people with pre-existing conditions,” said Sen. Chris Murphy (D., Conn.).

Organizations supporting the ACA plan to hold rallies and events in key battleground states such as Alaska, Arizona, Nevada, and Ohio, hoping to mirror the strategy used to block the ACA repeal last year.

Millions will be spent on television and digital ads, often featuring people with pre-existing conditions, according to Brad Woodhouse, campaign director at the advocacy group Protect Our Care. “Our strategy is to create a lot of sturm und drang in the Republican caucus,” he said.

GOP strategists say the Democrats’ arguments on the ACA are irrelevant to Judge Kavanaugh’s qualifications and will fail. Liberals, they say, are trying to change the subject from a debate on the nominee’s merits—which they would lose—to terrain they hope is more favorable.

“It’s no secret that Democrats think that health care is a winning issue for them, in part because Republicans have so far failed to develop a post- ‘repeal and replace’ message,” said GOP strategist Alex Conant. “Democrats will look for any opportunity to hammer that message, even if it’s tangential to the actual merits of the Supreme Court nominee.”

The Republican strategy is to emphasize Judge Kavanaugh’s background as a respected jurist, his reputation for a calm temperament, and praise he has elicited from some liberals.

On the health-care issue, Democrats are concerned about how the Supreme Court with a Justice Kavanaugh on it would rule in a federal lawsuit filed last year in Texas by 20 Republican state attorneys general arguing that the ACA is unconstitutional, since Congress recently repealed a provision that imposes a tax penalty on individuals without health insurance. The Supreme Court, in upholding the ACA, cited Congress’ taxing power, but without that tax provision it’s no longer constitutional, the suit argues.

The Trump administration filed a brief asking the court to strike down key elements of the law, including its ban on insurers denying coverage to people with pre-existing health conditions.

Still, the justice whose seat Judge Kavanaugh would be taking, Anthony Kennedy, dissented in the court opinion upholding the ACA. Judge Kavanaugh’s arrival wouldn’t necessarily change the level of support for the health law among the justices.

In any case, legal analysts say it’s far from certain the Texas lawsuit will reach the Supreme Court. It’s more likely, they said, that the court will see cases challenging individual actions the Trump administration has taken to roll back parts of the ACA.

“Those actions could end up before the Supreme Court,” said Nicholas Bagley, a University of Michigan law professor. “That’s where his influence will matter.”

Republicans say such Democratic attacks are scare tactics from those eager to derail a conservative nominee.

“My Democratic colleagues are going to throw everything they have at Judge Kavanaugh,” Sen. Orrin Hatch (R., Utah), a longtime Senate Judiciary Committee member, said recently on the Senate floor. “We’re going to see Judge Kavanaugh’s opponents twist his words, misrepresent his opinions, and do everything they can to make him into some sort of monster.”

In truth, Mr. Hatch said, Judge Kavanaugh is respected, levelheaded jurist. “He interprets the Constitution as written,” Mr. Hatch said. “He interprets our laws as written.”

Still, Mr. Hatch agreed the high court was likely to play a role in determining the ACA’s fate. “Anybody who thinks it’s not going to be litigated sometime in the future is nuts,” he said. “So whoever is on the court is probably going to see that type of matter come before the court and you can’t shy away from it.

For their part, some conservatives are uneasy with a 2011 ruling in which they believe Judge Kavanaugh didn’t go far enough in declaring the ACA unconstitutional. Instead, he found that a case challenging the law was premature because its provisions hadn’t taken full effect.

Sen. Joe Manchin of West Virginia, one of three Senate Democrats to vote for Mr. Trump’s first pick for the Supreme Court, Neil Gorsuch, said he is weighing the health-care issue as part of his deliberations. About 800,000 people in his state would be affected if protections for pre-existing conditions are dropped, he said.

“It’s a big concern for West Virginia,” Mr. Manchin said.