Just days after George Floyd’s death, a court officer in New York posted two pictures side-by-side on his Facebook page.

The first image was already instantly recognizable: Mr. Floyd, face down, with a police officer kneeling on his neck. The second showed the N.F.L. player Colin Kaepernick, kneeling to bring attention to instances of police brutality against Black people, among other forms of racism.

“What kneeling one offends you more?” a prewritten caption read.

“Clearly the football player,” the court officer wrote in response. “I’m sure the suspect deserved what he got.”

The officer, who is from the Buffalo area and has been working for the court system for almost a decade, was not fired after another employee brought the post to the attention of the inspector general’s office. He was not ordered to undergo a disciplinary hearing or required to take sensitivity training. Although he was suspended for a month, he has since returned to the job.

This week, New York’s chief judge, Janet DiFiore, said that settling the matter without a more thorough accounting had been a “mistake.” In a memo to the roughly 1,350 judges and 14,000 other employees of the state court, she wrote that such settlements “inevitably fosters a perception that such conduct, though penalized, is tolerated within our ranks. It is not.”

Judge DiFiore said, going forward, any case involving claims of discriminatory conduct would necessitate a full disciplinary hearing, the system’s equivalent of a trial. The officials who had been involved in the decision to settle the officer’s case would be “participating in immediate anti-bias training,” the judge said.

A court spokesman, Lucian Chalfen, said that Judge DiFiore and the state’s chief administrative judge, Lawrence Marks, had not known how the case had been handled until after it was reported on by the New York Law Journal in late January. “They were extremely displeased about it,” he said.

Mr. Chalfen said that he was barred from revealing the identity of the officer who wrote the racist post. But he said Judge DiFiore’s learning of the lenient measures taken against him had led to her memo.

The Civil Service Employees Association, the union which represents the officer, would not comment on the specifics of the case, other than to say in a statement that it did not “condone discrimination of any form, and we completely agree with the overall premise that workers deserve a court system free of any discrimination or bias.”

It said that the change proposed in the Judge’s memo “clearly diminishes worker rights and may violate our contract.” It also said that it was investigating whether the proposed change was illegal.

The New York court system’s tolerance for racist conduct from its personnel has been a subject of heightened scrutiny since Mr. Floyd’s death. On June 9, Judge DiFiore, spurred by Mr. Floyd’s death, commissioned a report to look into institutional racism in the court system. The team tasked with the report was led by Jeh C. Johnson, a former Homeland Security secretary under President Obama.

Mr. Johnson’s team found that racism was deeply embedded within the culture of New York’s court officers, the state’s term for the officials who keep order in the courts. (In some states, they are known as bailiffs.)

The team found specific instances of racist posts to social media, including one white officer who had posted a drawing of former President Barack Obama with a noose around his neck and another who referred to a Black colleague as “one of the good monkeys.”

“The use of racial slurs by white court officers is common and often goes unpunished,” the report also said.

Court officers of color told Mr. Johnson’s team “they felt they could not report incidents of bias for fear of being ostracized by their fellow officers and facing adverse career consequences from powerful and entrenched union leaders.” Union leadership itself, they said, was a “safe haven for racist speech and actions.”

Law enforcement agencies throughout New York have had to contend with racism in their ranks this year. Earlier this week, a top New York police official was fired after an internal investigation found that he had posted racist rants targeting Black, Hispanic and Jewish people. (The official, Deputy Inspector James F. Kobel, whose role it was to combat workplace harassment within the department, has denied any wrongdoing.)

Judge DiFiore addressed the larger racism problem in the courts in her memo this week.

Racism and other offensive sentiments, the memo said, had an impact upon the courts that was “immediate, irreparable and completely unacceptable.”

“Our approach to such behavior must be ‘zero tolerance,’” Judge DiFiore wrote.