“It is a mistake to think of this as a partisan fight over health care,” said Mr. Eggleston, who called the litigation critical to separation-of-powers issues. “The White House has Leader McConnell and Speaker Ryan now, but could easily be dealing before the end of the administration with Leader Schumer and Speaker Pelosi.”

He was referring, of course, to Senator Mitch McConnell of Kentucky, the majority leader, and Speaker Paul D. Ryan of Wisconsin — two Republicans unlikely to initiate a lawsuit against a Republican president — and Senator Chuck Schumer of New York and Representative Nancy Pelosi of California, two Democrats now leading the minority, with hopes of claiming the majority in 2018.

The 2016 decision by Judge Rosemary M. Collyer that the health care spending was unconstitutional and that the House could pursue the lawsuit was at only the district court level, and its reach was viewed as limited. The judge also said the unique circumstances of a case testing the authority of the congressional power of the purse — its most fundamental constitutional role — had led her to grant standing, suggesting she did not mean to bestow expansive new legal rights on Congress.

Still, the decision alarmed the White House sufficiently that it rapidly pushed forward with an appeal both to preserve its independence and to protect the health care law.

The decision by the Justice Department to drop its defense of the Obama program and cut off the subsidies was not unexpected. Since President Trump took office, his administration has shown little interest in taking up the Obama administration cause. Quite the contrary. Given their strong opposition to the Obama health care program, the Trump administration and House Republicans asked the appeals court to hold off on any action in the lawsuit until Congress decided what to do about the health care law. Unable to force through a repeal, the administration has now turned to other actions it can take, with the decision to cut off the subsidies just the latest one.

How the court case will ultimately be disposed of remains unclear. Nicholas Bagley, a University of Michigan law professor who has followed it closely, expects that the Justice Department will move to have it declared moot, since the main point of contention — the legality of the subsidies — has been settled by the new finding by Mr. Sessions.

“There is nothing left to fight about,” Mr. Bagley said.

It might not be so simple. A coalition of states successfully intervened in the lawsuit this year in an effort to protect consumers when it became clear that the Trump administration was not inclined to contest the House claim. Mr. Bagley said the states could ask the appeals court to block the Trump administration from cutting off the funding. In addition, insurance companies could go to court to recover the subsidies they had been promised under the law. There are also bipartisan efforts in Congress to formally appropriate the money.

One thing is clear: With the House and the executive branch now on the same legal side, there is no longer any separation between the two main powers in the case.