On June 9, the leadership of the Smithsonian gathered for a quarterly, but hardly routine, meeting behind closed doors.

President Trump had already called out the Smithsonian for being part of a “concerted and widespread effort to rewrite our nation’s history” and announced he was firing the head of its National Portrait Gallery.

Now the Smithsonian’s board planned to discuss a response — a resolution carefully calibrated to avoid a confrontation with the president. The resolution would reinforce that only the Smithsonian had the power to fire its museum leader, but would also order a full review of Smithsonian content for bias.

After the resolution had been introduced, Representative Carlos Gimenez of Florida, a Republican board member, interrupted, proposing instead that the board fire the gallery director, as Mr. Trump had sought. His effort was quickly shut down by the Smithsonian’s chancellor — the chief justice of the Supreme Court, John G. Roberts Jr.

“We already have a motion on the floor,” Chief Justice Roberts said, according to three people with knowledge of the proceedings.

The original resolution succeeded. The meeting quickly moved on.

If the moment was unusually tense for a gathering of a museum board, the intervention by the chief justice, a committed parliamentarian, was not. As chancellor, he is known to preside over meetings with a strict focus on rules and procedures, assiduously avoiding partisan debates — a demeanor that aligns with his reputation as an institutionalist and incrementalist jurist.