The Hollywood Reporter said the lawsuit seeks unspecified damages for JoDon Romero's sons, who are ages 9, 13 and 15.
Court papers allege the children have suffered signs of post-traumatic stress disorder since watching footage of their father shooting himself, following a police car chase and standoff.
Fox host Shepard Smith, who had been narrating the coverage, shouted at the control room: "Get off, get off, get off, get off, get off, get off, get off -- get off it, get off it, get off it," however, images of Romero killing himself aired live anyway.
Smith returned after a commercial break and admitted the network had "really messed up and we're all very sorry."
Fox later released a statement explaining "severe human error" was to blame for the disturbing video, which did not subsequently air on Fox News programs or on newscasts by Fox affiliates, including a station in Phoenix.
Although it didn't air in Romero's home state, his two older sons saw the video on YouTube where it was posted minutes after it was initially broadcast, the lawsuit said.
Fox News told The Hollywood Reporter it won't comment on pending litigation.