This question is partly about what laws congress can pass by simple majority and partly about the US constitution.
Could congress pass a law by simple majority that barred an individual from standing for presidential election in the future?
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The constitution only mentions who is disqualified. It doesn't say who is qualified. That is down to the electors to decide.
No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States.
And those who have served two terms may not stand again
No person shall be elected to the office of the President more than twice
It is implied that those convicted in a Senate trial may be disqualfied
Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States
Moreover, there is a particular amendment for automatic disqualification for rebellion (this is a civil war amendment):
No person shall be a Senator or Representative in Congress, or elector of President and Vice-President, or hold any office, civil or military, under the United States, or under any State, who, having previously taken an oath, as a member of Congress, or as an officer of the United States, or as a member of any State legislature, or as an executive or judicial officer of any State, to support the Constitution of the United States, shall have engaged in insurrection or rebellion against the same, or given aid or comfort to the enemies thereof. But Congress may by a vote of two-thirds of each House, remove such disability.
And it states that there can be no religious test for qualification:
[N]o religious Test shall ever be required as a Qualification to any Office or public Trust under the United States.
And a bill that states that "This person, or group of people committed a crime" is not allowed:
No Bill of Attainder or ex post facto Law shall be passed.
Now the route to become president is clear.
The person having the greatest number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed.
How is this interpreted? There is no clear congressional power to disqualify someone who has received a majority of votes in the electoral college, except in the case that that person has been convicted in a Senate trial, or otherwise disqualified, for example for criminal rebellion (which would seem to require conviction in court and be "beyond reasonable doubt"). And Congress does not have powers beyond those present or implied in the constitution. As the power to disqualify is not implied, except in the case of impeachment.
How is this interpreted — The understanding of this is that a person must first be convicted, and then disqualified. There is no process for ad hoc disqualification. The trial must come first. In the case of rebellion, the bar is high: The criminal standard of proof beyond reasonable doubt.
As mentioned here:
In an opinion piece published in The Washington Post on Monday, Yale Law School professor Bruce Ackerman and Indiana University law professor Gerard Magliocca argued that members of Congress have another, perhaps easier, path to barring Trump from office.
They pointed to the Section 3 of the 14th Amendment, aimed at preventing people from holding federal office if they are deemed to have “engaged in insurrection or rebellion against” the Constitution.
The professors write that if a majority vote of both houses agree that Trump engaged in an act of “insurrection or rebellion,” then he would be barred from running for the White House again. Only a two-thirds vote of each house of Congress in the future could undo that result.
The sole article of impeachment adopted Wednesday cites that provision of the Constitution and says Trump should be disqualified from holding future office.
WHAT’S SECTION 3 DOING IN THE 14TH AMENDMENT?
The 14th Amendment was one of three amendments adopted after the Civil War to end slavery and afford equal rights to Black people. The point of Section 3, according to Ackerman and Magliocca, was to keep Confederates — those who had engaged in “insurrection or rebellion” — from holding public office in the postwar period. In 1872, Congress passed the Amnesty Act to allow those men to serve again. But Section 3 remains. It was last used a century ago to keep a socialist from Wisconsin who opposed U.S. entry into World War I from taking his seat in Congress.
The socialist mentioned here was Victor L. Berger:
Berger's views on World War I were complicated by the Socialist view and the difficulties surrounding his German heritage. However, he did support his party's stance against the war. When the United States entered the war and passed the Espionage Act of 1917, Berger's continued opposition made him a target. He and four other Socialists were indicted under the Espionage Act in February 1918; the trial followed on December 9 of that year, and on February 20, 1919, Berger was convicted and sentenced to 20 years in federal prison.
During the 1918 Wisconsin special senate election, Berger ran for the seat under federal indictment. His newspaper, the Milwaukee Leader, had printed a number of anti-war articles which led to the postal service revoking the paper's second-class mail privileges. Despite the circumstances, Berger won 26% of the vote statewide in an April special Senate election to fill a vacancy, winning 11 counties in a three-way race.
The espionage trial was presided over by Judge Kenesaw Mountain Landis, who later became the first commissioner of Major League Baseball. His conviction was appealed, and ultimately overturned by the Supreme Court on January 31, 1921, which found that Judge Landis had improperly presided over the case after the filing of an affidavit of prejudice.
In spite of his being under indictment at the time, the voters of Milwaukee elected Berger to the House of Representatives in 1918. When he arrived in Washington to claim his seat, Congress formed a special committee to determine whether a convicted felon and war opponent should be seated as a member of Congress. On November 10, 1919, they concluded that he should not, and declared the seat vacant. He was disqualified pursuant to Section 3 of the Fourteenth Amendment to the United States Constitution. Wisconsin promptly held a special election to fill the vacant seat, and on December 19, 1919, elected Berger a second time. On January 10, 1920, the House again refused to seat him, and the seat remained vacant until 1921, when Republican William H. Stafford claimed the seat after defeating Berger in the 1920 general election.