In his 66-page ruling, the arbitrator accused several Fox senior executives — including the 21st Century Fox president Peter Rice and the Fox Television Group co-chief executive Dana Walden — of perjuring themselves. Mr. Lichtman said that those executives “appear to have given false testimony in an attempt to conceal their wrongful acts” and accused them of a “pattern of deceit.”

“The arbitrator is convinced that perjury was committed by the Fox witnesses,” Mr. Lichtman wrote.

Arbitration hearings are meant to be private and rarely come to light. Fox executives, however, were stunned by the ruling, which they saw as heavily one-sided, according to two people with direct knowledge of their thinking. The opinion left the company little leverage to negotiate a settlement, and the executives decided to contest the damages, the people said. That would also require public scrutiny before a judge.

In a statement, 21st Century Fox said, “The ruling by this private arbitrator is categorically wrong on the merits and exceeded his arbitration powers. Fox will not allow this flagrant injustice, riddled with errors and gratuitous character attacks, to stand and will vigorously challenge the ruling in a court of law.”

“Bones,” an hourlong crime show, was a sturdy hit for Fox from 2005 to 2017. It starred Emily Deschanel and David Boreanaz, both of whom were plaintiffs in the case, along with the executive producer Barry Josephson and the writer Kathy Reichs, who wrote the novels the series was based on.

An arbitration hearing requires both sides to agree to the terms of possible damages beforehand. The plaintiffs and defendants also agree in advance to abide by the ruling, without the recourse of an appeal, as in a court trial. But a party can contest an arbitration judgment if it determines that an arbitrator has awarded damages that fall outside the agreed upon terms.