If there were any doubts about the challenges facing House Democrats seeking to forge unity, just ask the 200 or so protesters who swarmed House Minority Leader Nancy Pelosi’s office to demand action on climate change.

Bolstered by an appearance by congresswoman-elect Alexandria Ocasio-Cortez, the 29-year-old political newcomer who beat a 10-term incumbent for a House seat from New York, the protesters staged a sit-in in the Democratic leader’s office in the Cannon House Office Building. Their demands included formulating...