The president has sought for months to end federal subsidies for insurance markets. And as recently as Friday, staunch conservatives have demanded the end of a special subsidy for House and Senate lawmakers and their staffs, through a District of Columbia insurance exchange, instead of a system specifically for federal employees.

In a statement on Saturday, Senator Chuck Schumer, Democrat of New York and the minority leader, said health care costs would rise for millions of Americans should the federal subsidy for insurance markets be scrapped.

“The president ought to stop playing politics with people’s lives and health care, start leading and finally begin acting presidential,” Mr. Schumer said.

Mr. Trump’s repeated criticisms of Senate process also have rankled Republican leaders.

Antonia Ferrier, a spokeswoman for Mr. McConnell, declined to comment on Mr. Trump’s posts. “If the leader issues any statements, we’ll be sure to pass along,” she said.

Mr. McConnell’s former chief of staff, Josh Holmes, cited Mr. Trump’s tweets on Saturday as he sardonically suggested on Twitter a “search for the idiot who keeps putting the President on irrelevant and counterproductive crusades.”