"He got out of the car and shot himself," Thompson said. "Efforts to revive him were not successful and he was dead at the scene. We don't have an ID yet."
Fox returned repeatedly to shots showing the car passing big-rig trucks that typically travel at about 70 mph as if they were standing still.
Police cars did not appear to be immediately behind the car during most of the chase.
Shepard Smith was narrating the video and clearly had his doubts about what was being shown from the moment the man stopped the car. "This scares me," he said.
"You wait for the end of these things and you worry about how they may end up," he said. "This makes me a little nervous, I got to tell you. A little nervous."
The video showed the man running erratically in a field before putting the gun to his head and firing. He fell to the ground.
Fox's picture quickly cut to Smith, who was shouting "get off, get off, get off, get off."
After the commercial break, Smith apologized repeatedly.
"That didn't belong on TV. We took every precaution we knew how to take to keep that from being on TV and I personally apologize to you that that happened," he said.
AP television writer David Bauder reported from New York. Associated Press Writer Walter Berry in Phoenix also contributed to this report.