Michelle Kivela, the public information officer for the city of Lone Tree, Colo., said Cox was arrested Thursday night following an investigation into an Oct. 28 incident and was transported to a nearby county jail.
Cox appeared Friday in Douglas County Court in nearby Castle Rock wearing a gray prison uniform. His feet were shackled and he was escorted by an armed sheriff's deputy.
A judge declined a defense attorney's request to reduce a $50,000 bond but did say that Cox can leave the state two times - apparently to play in Broncos away games. Cox waived extradition, meaning he agreed not to fight transport back to Colorado if for some reason he were arrested out of state.
The case filing was sealed at the request of the district attorney's office. Cox's next court hearing was set for Jan. 7.
Broncos media relations director Patrick Smyth said the team is "currently in the process of gathering information and reviewing the facts."
Cox has played in 11 games this season after being taken in the fifth round out of Oklahoma State.
Douglas County Court spokeswoman Casimir Spencer, prosecutor John Topolnicki and defense attorney Vince Buckmelter all declined to comment on the case.
Castle Rock is about 30 miles south of Denver.