Facing impeachment, the Homeland Security secretary, Alejandro Mayorkas, is refusing to slouch to the exit like previous cabinet members in his shoes. Where they resigned, he stands his ground — and in a nation wary of endless partisan vendettas, he likes his odds.

The chairman of the House Homeland Security Committee, Mark Green, wrote in a statement that Mr. Mayorkas “has willfully and systemically refused to comply with immigration laws enacted by Congress” and “breached the public trust.” The impeachment vote is set for Tuesday. 

In the two centuries since President Washington formed the first cabinet, multiple secretaries have faced impeachment inquiries. Resignation was a way to spare the administration they served embarrassment, but Mr. Mayorkas has taken the opposite road.

“I assure you,” Mr. Mayorkas wrote to House Republicans “that your false accusations do not rattle me.” That doesn’t sound like a man packing his bags, even if offered a golden parachute like President Hoover’s Treasury secretary, Andrew Mellon.

In the early years of the Great Depression, Hoover’s Treasury chief faced accusations of what the Federal Reserve describes as “improper business ties.” In January 1932, ahead of a House Judiciary Committee vote on impeachment articles, Mellon resigned. Hoover appointed him ambassador to the United Kingdom, the most prestigious post in foreign service, something of a lateral move.

Mellon, a Republican, risked impeachment in a House where Democrats held a six-seat majority. Mr. Mayorkas faces an opposition majority half that size, and there’s nothing like the Great Depression that destroyed Mellon’s reputation. 

Republicans seek to hold Mr. Mayorkas responsible for a crisis on his watch: The flood of humanity across the southern border. But they haven’t succeeded in making him the face of the problem the way Democrats did with Mellon.

In 1922, a cloud gathered over President Harding’s attorney general, Harry Daugherty, for allegedly taking kickbacks during Prohibition. Holding a whopping 172-seat majority, his fellow Republicans didn’t pass articles of impeachment and a Senate investigation found him not guilty, but he wasn’t safe.

By 1924, after Harding’s death, President Coolidge was in office. Hoover, then secretary of commerce, wrote in his memoir that he and the secretary of state, Charles Hughes, “urged that Daugherty had lost the confidence of the whole country and himself should be willing to retire for the good of public service.”

Hoover said that Coolidge was reluctant to demand Daughtery’s resignation based on “rumors,” but agreed to do so. In the end, it was a loss of trust, not impeachment, that doomed Daughterty. Mr. Mayorkas, however, has Mr. Biden’s backing and nothing like the broad public anger against bootleggers.

President Grant’s secretary of war, William Belknap, was the only Cabinet secretary ever impeached. Even he attempted to resign, doing so just 40 minutes before a committee was set to vote articles against him to the full House.

Belknap was accused of selling appointments, charges testified to by General George Custer. His case may present the closest allegory to Mr. Mayorkas, with partisans seeking to extract a pound of flesh from a president they opposed.

Democrats wanted to punish Grant for using the Army and Justice Department to crush their allies in the Ku Klux Klan. They settled for a scalp in the Republican’s Cabinet. But unlike the case against Mr. Mayorkas, House investigators in 1876 uncovered smoking guns. 

By March 5, 1876, the Sun reported “Belknap’s downfall” had exposed “fraud” and a “service honeycombed with corruption,” including “in the contracts for headstones” of Civil War dead. Belknap was later indicted, but Grant ordered the case dismissed, which the Sun praised as a “kindness.” 

The political divisions of Gilded Age America were akin to today’s, with narrow congressional majorities and two presidents who lost the popular vote triumphing in the Electoral College. Impeachment investigations have become more common as parties seek to break that stalemate. 

But the Constitution requires a two-thirds Senate majority for conviction. It’s a high bar that the Democratic-controlled upper chamber is unlikely to jump for Mr. Mayorkas, even absent his gaining mercy by resigning like Belknap.

Based on the history of executive branch impeachments, bet on Mr. Mayorkas to stick in his job until he or Mr. Biden decides it’s time to head for the exit — or Americans show them the door in November’s election.