Senate Majority Leader Chuck Schumer on Tuesday criticized U.S. Supreme Court Chief Justice John Roberts for what he sees as lackluster efforts to address ethical impropriety among high court members.

Justice Roberts has "not lived up to his responsibilities as chief justice," Schumer, D-N.Y., said at a press conference when asked whether or when the Senate would take up the high court ethics bill that last July passed the Senate Judiciary Committee along party lines.

Schumer's remarks come on the heels of recent press reports that two flags associated with participants in the Jan. 6, 2021, insurrection at the U.S. Capitol were flown at the homes of Justice Samuel Alito. Justice Alito has said his wife, Martha-Ann Alito, was responsible for raising the flags.

The resulting criticism has led to renewed calls to codify an ethics regime for the high court, and for Justices Alito and Clarence Thomas—whose wife, Virginia "Ginni" Thomas, pressed Trump White House staff to overturn the results of the 2020 presidential election—to recuse themselves from Jan. 6-related cases.

Schumer also argued that Justice Roberts, in his role as the presiding officer of the Judicial Conference of the United States, has done too little to rein in "forum shopping" by litigants who file cases in specific federal district courts they believe will be sympathetic to their lawsuits.

"On issue after issue, he's supposed to be the guardian of the court's fairness and opinion, but on something near and dear to me, which is forum shopping, on ethics, on recusal," he's not done enough, Schumer said.

Representatives for the Supreme Court did not immediately respond to a request for comment late Tuesday.

Schumer said he's consulting with the Senate Judiciary Committee, including its chair, Dick Durbin, D-Ill., about "the best ways to move forward."

Last week, Justice Roberts declined a meeting with Durbin and Sen. Sheldon Whitehouse, D-R.I., chair of the Judiciary Committee's federal courts panel, to discuss ethics issues.

"Apart from ceremonial events, only on rare occasions in our nation's history has a sitting chief justice met with legislators, even in a public setting (such as a committee hearing)," Justice Roberts wrote in declining the invitation. "Moreover, the format proposed—a meeting with the leaders of only one party who have expressed an interest in matters currently pending before the court—simply underscores that participating in such a meeting would be inadvisable."

In April, Schumer and other Democrats introduced a bill to codify the Judicial Conference's newly updated policy calling for random case assignments to make sure that litigants can't pick the judge they want by going to a one-judge division of a district court. The bill has yet to be voted on.

When the high court released its first-ever code of conduct last fall, Schumer said the code is "important, but its lack of enforceability is a glaring omission."

"Congress must continue its efforts to hold the judiciary accountable," he said at the time.