WASHINGTON (Reuters) - Democrats intensified calls on Thursday for U.S. Supreme Court Justice Samuel Alito to step aside in two cases - one on Donald Trump's bid for immunity from prosecution and the other on a charge involving the Capitol attack - after a media report that another provocative flag flew outside a home of the jurist.

A flag bearing the phrase "Appeal to Heaven" that was carried by some Trump supporters during the Jan. 6, 2021, Capitol riot flew outside Alito's vacation home on Long Beach Island, New Jersey, in July and September of 2023, the New York Times reported on Wednesday, citing photographs and interviews.

The flag has come to symbolize hopes by some conservative activists for a more Christian-centered U.S. government.

The report came days after it was revealed that an upside-down American flag - another symbol displayed by the Capitol rioters - had hung outside Alito's Virginia home in the aftermath of the riot and the days before Democratic President Joe Biden's inauguration.

The inverted flag was adopted as a symbol by some Trump supporters, including some of the rioters who stormed the Capitol as Trump sought to overturn his 2020 election loss to Biden with false claims of widespread voting fraud and a "stolen" election.

In a social media post on Thursday, Hakeem Jeffries, the top Democrat in the Republican-controlled U.S. House of Representatives, wrote, "The American people deserve better than an insurrectionist sympathizer on the Supreme Court."

"This incident is yet another example of apparent ethical misconduct by a sitting justice, and it adds to the court's ongoing ethical crisis," Dick Durbin, the Democratic chair of the Senate Judiciary Committee, said on Wednesday evening. "For the good of our country and the court, Justice Alito must recuse himself immediately from cases related to the 2020 election and the January 6th insurrection."

Durbin also said that the latest flag flap will "further erode public faith in the court," and that conservative Chief Justice John Roberts "must see how this is damaging the court and immediately enact an enforceable code of conduct."

"How many MAGA battle flags does Alito need to fly for the court or the Judicial Conference to see there's a problem?" Democratic Senator Sheldon Whitehouse wrote on social media on Wednesday night, referring to the "Make America Great Again" slogan used by Trump and his allies.

The Judicial Conference is the federal judiciary's policymaking body.

Alito, a member of the court's 6-3 conservative majority, did not immediately respond to a request for comment on Thursday.

One of the two cases at the center of the recusal calls involves Trump's claim of presidential immunity from prosecution on federal criminal charges relating to his efforts to overturn the 2020 election results. The other involves a challenge by a Pennsylvania man to a federal criminal charge of obstruction that he faces for his involvement in the Jan. 6 attack. Trump faces the same charge in the election-related criminal case brought against him by Special Counsel Jack Smith.

(Reporting by Andrew Chung and John Kruzel; Editing by Will Dunham)