Choice approached Vick after the Eagles' 30-27 win at Cowboys Stadium and asked him to sign a glove. The MVP candidate granted the request.
"That's probably the first time that's ever happened to me," Vick said of an opposing player's request after a game. "But [Choice] is a great guy. It doesn't surprise me."
Choice, who rushed for 16 yards on seven carries and had a 17-yard reception in the loss, said he has known Vick for years. They became acquainted when Choice attended Georgia Tech and Vick played for the Atlanta Falcons.
The autograph, according to Choice, was for a young family member who is a big fan of Vick.
"My little nephew says he's his brother," Choice said Sunday night. "My little nephew is 2 years old, so I had to get one for him. I just admire somebody who comes from those circumstances and comes back and proves everybody wrong. I've got nothing but respect for him."
On Monday morning, however, Choice seemed to regret making the request, which was shown on NBC's national broadcast.
"I don't want anybody to think I was disrespectful," Choice told ESPNDallas.com. "My teammates, coaches and Jerry [Jones] know where my heart is. I care about football. I meant no disrespect."
Interim coach Jason Garrett said he intended to discussion the situation with Choice, although Garrett said he was not angry with the third-year running back.
Garrett staunchly defended Choice, emphasizing the running back's competitiveness and commitment to the Cowboys.
"I know how he prepares and how he feels about playing football and the passion with which he plays," Garrett said. "Anything beyond that, I don't really want to comment on, but he goes about it the right way and we're glad we have him on our side. We're glad he takes the approach that he does."
Several of Choice's teammates also downplayed the postgame autograph.
"It's not like he did it in the middle of the game," defensive end Stephen Bowen said. "I don't know why you would be upset. At the end of the day, when you're playing there is no friends, but at the end of the game they are all friends. I personally don't see it as a big deal."
"That did not affect the outcome of our game," said linebacker Keith Brooking, a team captain. "I'm worried about things that are going to affect the outcome of our game and help us win. Tashard getting an autograph after the game from a player didn't affect us losing."
Tim MacMahon covers the Cowboys for ESPN Dallas. You can follow him on Twitter. Calvin Watkins of ESPNDallas.com contributed to this report.