Senate Democrats are facing intensifying pressure from the left for inquiries into ethical questions at the Supreme Court, but they say their options are limited given the court’s independent status and Republican opposition.

Advocacy groups and progressives are stepping up their calls for Senate Democrats to be more aggressive after Chief Justice John G. Roberts Jr. rebuffed a plea to require Justice Samuel A. Alito Jr. to recuse himself from pending cases on the Jan. 6 assault on the Capitol and Donald J. Trump’s immunity for any actions leading up to it.

Top Democrats on the Judiciary Committee called on Justice Alito to recuse himself from those cases following reports that two flags associated with the “Stop the Steal” movement were flown outside his residences. The justice said his wife was responsible for the flags and refused to step away from the cases.

Earlier this week, a coalition of liberal groups and House Democrats urged the Senate to open an investigation into Justice Alito’s actions, with activists arguing that the Senate needed to stop behaving as if it were powerless.

Representatives Jamie Raskin of Maryland and Alexandria Ocasio-Cortez of New York, the top two Democrats on the Oversight Committee, have scheduled a round table on the issue for next Tuesday. It is set to explore a range of controversies surrounding the court, including unreported gifts and travel provided the justices, and their impact on the court’s agenda.

“The whole country is caught in a supreme ethics crisis,” Mr. Raskin said in a statement. “Our democracy, voting rights, and fundamental rights are on the line, and everything we have fought for is in danger because of this runaway court.”