But Mr. Murdoch is not just any chief executive, and the company he runs, especially its cable news network, has enormous influence over the country’s politics and media.
Rupert Murdoch, 86, who has long pursued power rather than a specific ideology, has served as an informal adviser to Mr. Trump and repeatedly urged him to fire Stephen K. Bannon, the chief strategist and nationalist who exited the White House on Friday.
At the same time, the Murdoch family controls both News Corporation, the owner of The Journal and The Post, as well as 21st Century Fox, home to a sprawling collection of movie studios and television networks. While there have been some cracks in the conservative wall, with the newspapers publishing more commentary critical of Mr. Trump, Fox News is known as Mr. Trump’s preferred outlet.
That creates an uneasy balance for James Murdoch, 44, who is known to lean more toward the center than his father, but rarely expresses political views publicly. A fiscal conservative, James and his progressive-minded wife, Kathryn, have long advocated for the environment and expressed embarrassment by certain elements of Fox News, associates have said. Kathryn Murdoch has expressed contempt for Mr. Trump on her Twitter feed.
The response from other Twitter users is often critical.
“Well Kathryn it would help if your family’s business #FoxNews wasn’t a synchophantic state media arm of the Trump regime #WednesdayWisdom,” one user said. “Some in your orbit have potential to alter the media ecosystem within which his parasitic organism self thrives,” said another.
Despite all that, the Murdoch sons have said repeatedly that they didn’t plan to significantly change the formula for Fox News, which fuels the company’s business. Analysts estimate that the division generated 25 percent of 21st Century Fox’s operating income last year, which was $6.6 billion.
“He is trying to straddle a recognition that there are a lot of problems out there, and whether Fox News has contributed to them or not, the problems exist,” said Brian Wieser, a media analyst with Pivotal Research. “Even though James is technically the C.E.O., he’s somewhere between can’t and won’t do anything that would cause changes to Fox News. This is a tricky divide.”
Mr. Wieser, who has a buy rating on 21st Century Fox, said that the most common pushback he received from investors involves their concerns about the future of Fox News, calling James Murdoch a “liberal” who will “ruin Fox News,” he said.
“James has to be mindful that the health of the overall enterprise is dependent on Fox News holding up,” Mr. Wieser said.
Through a spokesman, the Murdochs declined to comment.
In the email sent Thursday, Mr. Murdoch acknowledged that he rarely offers “running commentary on current affairs.”
Since he assumed the role of chief executive of 21st Century Fox two years ago, James Murdoch and his brother have pushed to modernize the company. They introduced additional benefits, including more paid vacation, vastly enhanced reproductive coverage for women and “expanded coverage for our transgender colleagues.” And in January, they spoke out about President Trump’s travel ban, stating in a memo to employees that they “deeply value diversity and believe immigration is an essential part of America’s strength.”
While there are some examples of the brothers moving to shake up the business, it is not clear how far they will go. In the face of the sexual harassment scandal at Fox News, it was James Murdoch who moved most aggressively against Roger Ailes, the founding chairman of Fox News, and Bill O’Reilly.
Still, some critics and employees said the Murdochs ousted Mr. Ailes and Mr. O’Reilly only because of public scrutiny, and that the work environment had not been completely reformed.
Some business associates and observers on Friday lauded Mr. Murdoch for taking a stand and making a donation to a charitable organization. Yet others questioned the sincerity behind his motives.
Angelo Carusone, president of Media Matters, a progressive media watchdog, said that Mr. Trump repeated several Fox News talking points during the news conference earlier this week when he responded to questions about what happened in Charlottesville.
“Much of what Donald Trump said this week that James Murdoch is condemning actually came directly from Fox News,” Mr. Carusone said. “So if James Murdoch really believes what he wrote in that email, then he needs to start with Fox News, the network that he runs.’’
It is not clear whether James Murdoch cleared the note with his father before sending it, although business associates said that it would be surprising if Rupert Murdoch hadn’t signed off on it.
The elder Mr. Murdoch has long supported the Anti-Defamation League, and accepted an award from the organization in 2010 for his “commitment to promoting respect and speaking out against anti-Semitism.”
For James Murdoch, his memo underscores that, at least for him, speaking out against racism, anti-Semitism, white supremacy and neo-Nazis exists outside political ideology.
“But what we watched this last week in Charlottesville and the reaction to it by the president of the United States concern all of us as Americans and free people,” Mr. Murdoch said in the email.
“The presence of hate in our society was appallingly laid bare as we watched swastikas brandished on the streets of Charlottesville and acts of brutal terrorism and violence perpetrated by a racist mob,” he added. “I can’t even believe I have to write this: Standing up to Nazis is essential; there are no good Nazis. Or Klansmen, or terrorists. Democrats, Republicans and others must all agree on this, and it compromises nothing for them to do so.”