The Democratic leader, Senator Chuck Schumer of New York, publicly scolded Mr. Burr, saying that he was “very disappointed.” The Republican’s behavior, Mr. Schumer said, “certainly gives the appearance, if not the reality, of a lack of impartiality.”

That rebuke aside, there is not much the Democrats can do to punish Mr. Burr. The committee’s first public hearing since the F.B.I. director disclosed that the bureau was investigating links between Trump campaign officials and Russia is scheduled for next week.

In coming weeks, the Senate committee would need to issue subpoenas to the Central Intelligence Agency and the Federal Bureau of Investigation for documents related to the inquiry. If Mr. Burr declines to issue them, Democrats would likely portray that as a Republican refusal to conduct an independent investigation, and accelerate demands for an outside commission.

Calls for an Independent Commission

Since shortly after the election, Democrats have pushed for the creation of an independent inquiry similar to the Warren Commission, which investigated the assassination of President John F. Kennedy, or the commission formed after the terror attacks of Sept. 11, 2001. Representative Eric Swalwell, Democrat of California, has introduced legislation to form the commission.

Under Mr. Swalwell’s proposal, Congress would appoint 12 members to investigate whether there was interference in the election, who was behind it and how it could be prevented in the future. The members of the commission would be pulled from a pool of former senior national security, law enforcement and election officials who are respected by both parties.

Among the type of officials who could be appointed to such a panel are Robert M. Gates, the former secretary of defense under President Barack Obama and President George W. Bush; Colin Powell, the former secretary of state and chairman of the Joint Chiefs of Staff; and Robert S. Mueller, the F.B.I. director under Presidents Bush and Obama. The commission would have a full-time investigative staff and the power to review classified information, subpoena witnesses and issue a public report.

Democrats had toyed with the idea of asking for the creation of a select committee that would be made up just of members of Congress — similar to the committee that investigated the 2012 attacks on a diplomatic compound in Benghazi, Libya.