Powell stopped Moats on March 18 outside Baylor Regional Medical Center in Plano, Texas, after he rolled through a red light.
Moats said he was rushing to see his dying mother-in-law, who was losing her battle with breast cancer. He says he rolled through the red light when it was safe to do so.
Powell is accused of pointing his gun at the couple and verbally abusing them during the traffic stop.
Moats' wife said she couldn't think about anything during the ordeal except getting to her mother's side.
"When you're in a situation like that, you really don't think about how dangerous it is because honestly, he could have shot me. He pointed the gun at me and he basically said don't take another step, don't move. Get back here," Tamishia Moats said.
Ryan Moats said he tried to reason with the officer, but that it did not work.
"When I got out, I thought if I explained it to him, maybe he would understand, so I went to explain it to him and I wasn't getting anywhere, and I said how about I be quiet and let him say what he has to say and maybe he'll just go ahead and write the ticket and let me go," he said.
Moats' mother-in-law died before he got inside.
Dallas Police Chief David Kunkle later commended Moats for not identifying himself as an NFL player or asking for special treatment.
In a written statement, Powell said, "I wish to publicly and sincerely apologize to the Moats family. After stopping Mr. Moats' vehicle, I showed poor judgment and insensitivity to Mr. Moats and his family by my words and actions. With great remorse, I accept my responsibility for adding to their grief in an already difficult time."
The Moats said they have not heard from Powell directly, but that they would accept his apology.
Powell, a three-year member of the force, remains on paid administrative leave pending an internal investigation.
The Associated Press contributed to this story.
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