The second-year player participated in the team's first practice of training camp Thursday.
Coming off the field, Cox told The Associated Press: "I just want to apologize for everything that's been going on. I'm truly grateful that football has started back up."
He acknowledged his legal troubles still weigh heavily on him.
"It's something I'm not going to let hold me back," Cox said. "Try not to think about it at all and try to keep pushing forward."
Cox was arrested last December and pleaded not guilty to one count of sexual assault while the victim was physically helpless and one count of sexual assault while the victim was incapable of determining the nature of the conduct. If convicted, Cox faces two years to life in prison.
For months, the arrest affidavit in the case remained closed as Cox's lawyer fought to keep it sealed.
But last week, the document was unsealed and it said Cox was linked by DNA to the alleged sexual assault victim.
Given the serious nature of the case, there was some speculation Cox might be let go by the Broncos once the lockout was lifted. Broncos President Joe Ellis said this week the team will monitor Cox's legal situation.
Cox is trying to keep his concentration on the field as he attempts to make the squad under new coach John Fox.
"A lot of people are talking about how I was going to be coming back," Cox said. "This is my job. It's all I'm focused on. I'm glad to be with my teammates, coaches.
"It's (nice) to be away from everything else, be around my teammates and play football."
He fully expects to be with the team when camp breaks and the season starts.
"I'll be here, I'll be here. No doubt," the 24-year-old said.
Cox still could face a four-game suspension from the NFL for violating the league's personal conduct code. Commissioner Roger Goodell doesn't need a player's conviction to dole out punishment.
On the first day of practice, Cox was running with the second team, backing up perennial Pro Bowler Champ Bailey and veteran Andre' Goodman. Cox doesn't know what his specific role might be this season. He could be a third defensive back in passing situations.
"I'll just go in whenever my name is called, try to make the best out of it," Cox said.
The former Oklahoma State standout was thrust into the starting role last year because of injuries. Opposing quarterbacks challenged him quite often and he showed flashes of his potential, batting away 14 passes and coming down with one interception. But he also made mistakes and gave up big plays.
Since his legal troubles surfaced, Cox said he's received plenty of support from inside the locker room.
"My teammates come up and say, 'Keep your head up, keep pushing forward,'" Cox said. "I'm here to play football."
The arrest affidavit revealed more details about Cox's arrest involving a woman who told officers she was likely drugged, raped and impregnated with the football player's child.
The woman told investigators that doctors put the date of conception around the time she went to Cox's apartment with friends in the early hours of Sept. 6, according to the affidavit.
The affidavit said DNA tests indicated Cox was the father. Cox is free on $50,000 bail.
Copyright 2011 by The Associated Press