The administration’s most pressing goal, expansive trade legislation, is adamantly opposed by scores of Democrats in the House and Senate even as most Republicans support it. Mr. Obama’s formal request for congressional authorization to fight the Islamic State is deeply imperiled, in no small measure because Senate Democrats find it wanting.

The president’s dismissal of the role of Congress in approving any nuclear agreement with Iran was facing a potential veto override before Republicans scrambled the political dynamic by sending a letter directly to the Iranian leadership. But if international negotiators reach an accord this month, Democrats’ concerns are all but certain to roar back.

Efforts to change a national security program that sweeps up Americans’ phone call data and other records have also stalled over disagreements with Democrats, even though both parties seek changes to the program.

Democrats may be unified on most of the policies that Mr. Obama called “middle-class economics” in his State of the Union address — which he has little chance of advancing with Republicans in control of Congress — but these other potential achievements may be imperiled.