Justice Samuel A. Alito Jr. declined on Wednesday to recuse himself from two cases arising from the Jan. 6, 2021, attack on the Capitol after reports that flags displayed outside his houses appeared to support the “Stop the Steal” movement.

In letters to Democratic members of Congress who had demanded his recusal, Justice Alito said that the flags, at his home in Virginia and a beach house in New Jersey, were flown by his wife, Martha-Ann.

“My wife is fond of flying flags,” the justice wrote. “I am not. She was solely responsible for having flagpoles put up at our residence and our vacation home and has flown a wide variety of flags over the years.”

The revelation that provocative flags flew outside the Alitos’ property has raised questions about the appearance of bias in two cases the Supreme Court is considering related to Jan. 6. In the weeks after the Capitol attack, an inverted American flag that Trump loyalists have adopted to contest Joseph R. Biden Jr.’s electoral victory was aloft at Justice Alito’s residence in Alexandria, Va. Last summer, an “Appeal to Heaven” flag, carried by rioters at the Capitol and now a symbol of support for a more Christian-minded government, was on display at his vacation house on Long Beach Island.

Former President Donald J. Trump, who is making another bid for the White House in the shadow of a series of criminal charges, welcomed Justice Alito’s decision to participate in the cases related to Jan. 6. One involves whether Mr. Trump is entitled to immunity from prosecution on charges that he plotted to subvert the election. The other concerns a federal obstruction statute used to charge hundreds of rioters who stormed the Capitol.

“All U.S. Judges, Justices, and Leaders should have such GRIT,” he wrote on his social media site.

In his letter on Wednesday, Justice Alito repeated his explanation for the upside-down flag while disclosing that his wife resisted his appeals to remove it after he learned of its existence.

“I had nothing whatsoever to do with the flying of the flag,” he wrote. “I was not even aware of the upside-down flag until it was called to my attention. As soon as I saw it, I asked my wife to take it down, but for several days, she refused.”

He said he had been powerless to remove the flag.

“My wife and I own our Virginia home jointly,” the justice wrote. “She therefore has the legal right to use the property as she sees fit, and there were no additional steps that I could have taken to have the flag taken down more promptly.”

On the other hand, Justice Alito wrote that the “Appeal to Heaven” flag at his New Jersey beach house did not convey the meaning critics ascribed to it. The flag, which shows a green pine tree against a white backdrop, dates back to the American Revolution and had mostly fallen into obscurity until a far-right Christian figure helped repopularize it in recent years.

“I was not aware of any connection between that historic flag and the ‘Stop the Steal Movement,’ and neither was my wife,” he wrote. “She did not fly it to associate herself with that or any other group, and the use of an old historic flag by a new group does not necessarily drain that flag of all other meanings.”

He added: “As I said in reference to the other flag event, my wife is an independently minded private citizen. She makes her own decisions, and I honor her right to do so.”

In sum, he said he would not disqualify himself from the two cases.

“A reasonable person who is not motivated by political or ideological considerations or a desire to affect the outcome of Supreme Court cases,” Justice Alito wrote, “would conclude this event does not meet the applicable standard for recusal.”

The court recently adopted a code of conduct for the justices, which Justice Alito said required him to participate in the cases. He quoted what he said were the relevant provisions.

One provision says that “a justice is presumed impartial and has an obligation to sit unless disqualified.”

A second provision says that “a justice should disqualify himself or herself in a proceeding in which the justice’s impartiality might reasonably be questioned, that is, where an unbiased and reasonable person who is aware of all relevant circumstances would doubt that the justice could fairly discharge his or her duties.”

The two incidents do not satisfy the second provision, he wrote, “and I therefore have an obligation to sit.”

Justice Alito said his wife was entitled to express her views, adding that the upside-down flag she had raised in Virginia was prompted by a heated dispute with neighbors. In his explanation, he offered some details of the dispute that were at odds with the account that the family involved and another neighbor told The New York Times, as well as text messages and a police record. The discrepancies include whether one of the encounters Justice Alito described precipitated the flag flying or came afterward.

“My wife is a private citizen, and she possesses the same First Amendment rights as every other American,” the justice wrote. “She makes her own decisions, and I have always respected her right to do so.”

“She has made many sacrifices to accommodate my service on the Supreme Court,” he wrote, “including the insult of having to endure numerous, loud, obscene and personally insulting protests in front of our home that continue to this day and now threaten to escalate.”

Justice Alito said his wife was exceptionally fond of flags.

“In addition to the American flag, she has flown other patriotic flags (including a favorite flag thanking veterans), college flags, flags supporting sports teams, state and local flags, flags of nations from which the ancestors of family members came, flags of places we have visited, seasonal flags and religious flags.”

Their beach house, on Long Beach Island in New Jersey, he wrote, was Mrs. Alito’s property.

“Our vacation home was purchased with money she inherited from her parents and is titled in her name,” he wrote. “It is a place, away from Washington, where she should be able to relax.”