Democrats, seizing on the health law’s popularity and its decisive role in their winning the House last fall, are already using the case as a cudgel against President Trump as his re-election campaign gets started. The law’s guarantee of coverage for people with pre-existing medical conditions, in particular, remains very popular with voters in both parties as well as independents.

But Mr. Trump has appeared undaunted, tweeting in April that “Republicans will always support Pre-Existing Conditions” and that a replacement plan “will be on full display during the Election as a much better & less expensive alternative to Obamacare.”

Instead of providing specifics, though, Mr. Trump, members of his administration and other Republicans have focused on attacking the Medicare for All plans that some Democratic presidential candidates have sponsored or endorsed as a dangerous far-left idea that would, as Mr. Trump tweeted, cause millions of Americans “to lose their beloved private health insurance.”

As the administration and Texas noted in their briefs, Judge O’Connor’s ruling turned on the law’s requirement that most people have health coverage or be subject to a tax penalty.

But in the 2017 tax legislation, Congress reduced that penalty to zero, effectively eliminating it. Judge O’Connor, the plaintiff states, and now the Trump administration reasoned that, like a house of cards, when the tax penalty fell, the so-called individual mandate became unconstitutional and unenforceable. Therefore, the entire law had to fall as well.

Mr. Paxton, the Texas attorney general, whose office also filed a brief on Wednesday, said: “Congress meant for the individual mandate to be the centerpiece of Obamacare. Without the constitutional justification for the centerpiece, the law must go down.”

Whether that position will survive judicial scrutiny is another question. Nicholas Bagley, who teaches health law at the University of Michigan Law School, noted that only two lawyers signed the brief. That is highly unusual in a case with such a high profile, he said.