Washington (CNN) President Donald Trump has been adamant about one point this week: "The Republican Party will soon be known as the party of health care."

How exactly that will come to be -- and whether it will be a blessing or a curse for Republicans in 2020 -- is still unclear.

Health care -- and Trump's attempts to roll back Obamacare -- became a galvanizing issue for Democrats in the 2018 midterms. Four in ten voters in 2018 said health care was the most important issue facing the country, according to a CNN exit poll, as they swept a wave of Democratic lawmakers into office to give Democrats the majority in the House of Representatives.

And now that they have lost the majority, Republicans have virtually no chance of passing any legislation that would repeal and replace President Barack Obama's signature health care law.

But Trump has been obstinate in the face of those political warning signs, instead concerned that repealing and replacing Obamacare remains the one signature 2016 campaign promise he has yet to make good on.

It's with that in mind that Trump backed his White House chief of staff Mick Mulvaney's arguments that the administration should support a federal judge's ruling in Texas invalidating Obamacare in its entirety, a senior White House official and Republican close to the campaign told CNN.

"Right now in the Texas court -- probably ends up in the Supreme Court," Trump told reporters on Friday. "But we're doing something right now that will be much less expensive than Obamacare for the people."

But what exactly that is remains a work in progress.

The uncertainty hasn't stopped Trump from tying himself and his party to the issue.

A week of health care moves

After the administration kicked off the week with its surprise shift to supporting the invalidation of the Affordable Care Act in the Texas case, Trump has been beating the drum constantly, thrusting the thorny issue into the center of the political arena even as federal judges in other cases ruled against him. On Wednesday, one jettisoned the administration's approval of Medicaid work requirements in two states and another on Thursday blocked a rule that would make it easier for small businesses to band together to buy health insurance, which could undermine Obamacare.

Trump's efforts have prompted grumbles -- private and public -- from Republican lawmakers mindful of the uphill battle Trump is setting them up for.

Senate Majority Leader Mitch McConnell on Thursday sought to distance his caucus from Trump's latest venture, saying he looks "forward to seeing what the President is proposing and what he can work out with the (House) speaker."

The President, they said, wants to telegraph to his political base that he is standing firm on his promise to repeal the entire law despite failing to wrangle Republicans to do just that during a drawn out legislative battle in 2017, when the party held both chambers of Congress. And he is now pinning his hopes on the courts accomplishing what he and a Republican majority in the House and Senate could not.

"We have a chance of killing Obamacare. We almost did it but somebody unfortunately surprised us with thumbs down, but we'll do it a different way. You know what? We'll do it a different way," Trump told his supporters at a rally in Grand Rapids, Michigan on Thursday night. "So we're going to get rid of Obamacare and I said it the other day: The Republican Party will become the party of great health care. It's good. It's important."

Trump's decision to support a ruling that would fully dismantle Obamacare, even as his administration has no viable alternative, is a risky gamble. If the ruling is upheld by higher courts, about 20 million Americans could lose healthcare coverage on Trump's watch.

It was that concern -- and qualms about the legal justification for the move -- that drew objections from Health and Human Services Secretary Alex Azar, Attorney General William Barr and White House counsel Pat Cipollone as the White House debated its position on the Texas lawsuit, which was brought by a coalition of Republican-led states. The Texas ruling is being appealed by a coalition of Democratic-governed states, because the Trump administration declined to defend the law in the first place.

Presidential working group

The President said Thursday he had tasked a handful of Republican senators with cobbling together an Obamacare replacement. No such working group appears to exist, however.

"I think the President just listed off the names of people he's spoken to on the phone about health care," one Senate Republican aide told CNN.

Spokesmen for Sens. Bill Cassidy of Louisiana and Rick Scott of Florida, two of the Republican senators Trump listed as being part of the group, said the senators have continued to have conversations with the President and their colleagues about healthcare issues, but made no mention of a working group.

Sen. John Barrasso of Wyoming, the third senator Trump name-checked, spoke with Trump about health care on Wednesday morning, but his staff said they did not have anything to share at the moment when asked about the specific working group.

The President's comments about the senators amounted to the latest attempt by the White House to reassure the public that it will have a replacement plan ready by the time the courts render a final judgment.

Just a day earlier, Marc Short, Vice President Mike Pence's chief of staff and the former White House legislative affairs director, claimed on CNN that "the President will be putting forward plans this year" to replace Obamacare through Congress.

White House officials were quick to tell CNN that Short had gotten ahead of White House deliberations.

The White House has yet to decide whether it will take the lead on crafting an Obamacare replacement, they said, or whether the President will punt to Republican lawmakers.