Officer Robert Powell also drew his gun during the March 18 incident involving Houston Texans running back Ryan Moats in the Dallas suburb of Plano, police said.
"I can screw you over," he said at one point in the videotaped incident. When another officer came with word that Moats' mother-in-law was indeed dying, Powell's response was: "All right. I'm almost done."
Dallas Police Chief David Kunkle apologized to the family and announced that Powell would be on paid leave pending an internal investigation.
"When we at the command staff reviewed the tape, we were embarrassed, disappointed," Kunkle said. "It's hard to find the right word and still be professional in my role as the police chief. But the behavior was not appropriate." CBS 11: Watch apology.
Powell, 25, a three-year member of the force, stopped Moats' SUV outside Baylor Regional Medical Center at Plano after Moats rolled through a red light.
Police officials said Powell told his commanders he believed he was doing his job, and that he drew his gun but did not point it. Kunkle said Powell was not necessarily acting improperly when he pulled his weapon out, but that once he realized what was happening should have put the gun back, apologized and offered to help the family in any way.
"His behavior, in my opinion, did not exhibit the common sense, the discretion, the compassion that we expect our officers to exhibit," Kunkle said.
Moats' wife, who was in the car along with other relatives, said Powell pointed his weapon at her.
"He was pointing a gun at me as soon as I got out of the car," Tamishia Moats told The Dallas Morning News.
Ryan Moats told KRLD-FM in Dallas in a phone interview Thursday that after the officer pointed the gun at his wife, he pointed it at him. "I just tried to stay as still as possible to not scare him or do anything to make him react," he said.
He earlier told the newspaper he thought Powell should be fired but backed off that in his radio interview.
"All I know is what he did was wrong," Moats said. "He stole a moment away from me that I can never get back. I'm really not the judge on what should happen to him."
The Moats family did not immediately return messages left by The Associated Press. Powell did not respond to requests for comment through the Dallas police union.
Video from a dashboard camera inside the officer's vehicle, obtained by Dallas-Fort Worth station WFAA-TV, revealed an intense exchange in which the officer threatened to jail Moats.
He ordered Tamishia Moats, 27, to get back in the SUV, but after pausing for a few seconds, she and another woman rushed into the hospital. She was by the side of her mother, 45-year-old Jonetta Collinsworth, when she died a short time later from breast cancer.
"Get in there," said Powell, yelling at Tamishia Moats as she exited the vehicle. "Let me see your hands!"
"Excuse me, my mom is dying," Tamishia Moats said. "Do you understand?"
Ryan Moats explained that he waited until there was no traffic before proceeding through the red light. When Powell asked for proof of insurance, Moats grew more agitated and told the officer to go find it.
"My mother-in-law is dying! Right now! You're wasting my time!" Moats yelled. "I don't understand why you can't understand that."
As they argued, the officer got irritated.
"Shut your mouth," the officer said. "You can either settle down and cooperate or I can just take you to jail for running a red light."
A nurse told officers Moats's mother in law really was dying, reports CBS News correspondent Mark Strassmann. Powell's response? "I'm almost done here."
By the time the 26-year-old NFL player received a ticket and a lecture from Powell, about 13 minutes had passed. When he and Collinsworth's father entered the hospital, they learned Collinsworth was dead.
Earl Jackson, Collinsworth's father, said he knew what Powell was doing was wrong. "This guy, he wouldn't listen to nobody," Jackson said in an interview with Dallas-Fort Worth station KDFW-TV.
Moats said he wouldn't have had a problem with the officer giving him a ticket after letting him go into the hospital.
"I don't know what he was thinking," he told KRLD-FM. "Basically, I was just shocked. I was very shocked that he wasn't budging on it. I even said I can't believe that this was happening."
Kunkle said the video showed that Moats and his wife "exercised extraordinary patience, restraint in dealing with the behavior of our officer."
"At no time did Mr. Moats identify himself as an NFL football player or expect any kind of special consideration," Kunkle said. "He handled himself very, very well."
The Moats family, who are black, said they can't help but think that race might have played a part in the white officer's behavior.
"I think he should lose his job," Ryan Moats said.
When the exchange was at its most contentious, Powell said he could tow Moats' SUV if he didn't have insurance and that he could arrest him for fleeing because he didn't immediately stop when Powell turned on his sirens. The pursuit lasted a little more than a minute.
"I can screw you over," Powell said. "I'd rather not do that. Your attitude will dictate everything that happens."
The ticket issued to Moats was dismissed, Dallas police spokesman Lt. Andy Harvey said.
Texans spokesman Kevin Cooper said the team had no comment.
Moats, a third-round draft choice of the Philadelphia Eagles in 2005 out of Louisiana Tech, was cut by the Eagles in August and later signed with the Texans. In three seasons as a backup, he's rushed for 441 yards and scored four touchdowns.
He was a standout at Bishop Lynch High School, a private school in Dallas, rushing for more than 2,600 yards and 33 touchdowns as a senior.