John Skipper was furious.

One of his star anchors, Jemele Hill, had sent a tweet calling President Donald Trump a "white supremacist." Mr. Trump's supporters called for her to be fired. Prominent black athletes defended the anchor, who is African-American.

Sitting in his office last September, Mr. Skipper, then ESPN's president, lit into Ms. Hill, according to people familiar with the meeting. If I punish you, he told her, I'd open us up to protests and come off as racist. If I do nothing, that will fuel a narrative among conservatives—and a faction within ESPN—that the network had become too liberal.