Yes, I think that character matters in professional sports and that all professional athletes need to be on their best behavior. Too many kids and young professionals look up to those who have been playing professional sports for awhile and they are on television and the Internet, which means that young kids are vulnerable.
One of the most important factors of sports is a fair game. Playing with humility is the best way to tackle the opponent. All the championships of the world have 'fair play' as one of the mottos of the sport. Also, if there is no character in pro sports, it would result in sheer barbarianism, where the only objective will be to win, not to play fair. Atheletes would do anything, right from sabotaging the opponents, to even taking his life! Also, the other example of characterlessness-using drugs to win- will reign supreme. Think about this- the competition will not be about who has the best ability, but about who uses better drugs! Pro sports promote life skills like humbleness in victory, team spirit, not backing down even after defeat, and most important of all, having character in life, like atheletes should have in sports.
To succeed in professional sports you need to be very disciplined in your training at some point in your life. None who train to be an athlete can do so when skipping sessions for example.
The moral sort of character needed to behave nicely to other people and be sensitive to their feelings is an entirely different cup of tea. There are no immediate repercussions on someone that is very proficient and puts your her head in the dirt and does the work that is required to be a competitor making the lineup of a professional team.
So let us not seek what we view as moderate behavior in people that grind soccer head shots, basket 3-point throws, hockey far lounges or other specialized skills with great character.
If your not willing to get a little scratched up or get ready to fight you shouldn't be playing sports. If you're playing professional football you should be prepared to get hurt, or get into a fight or two. If you get hurt no one is going to care because when you told yourself you wanted to play football you knew the risk. You knew the risk because YOU SIGNED UP FOR THIS!
I believe that character is incredible important in sports, however for the sake of argument, I will play the devil's advocate. A large difference from playing in a professional league (such as the NFL), vs playing in a non-professional (peewee/highschool, etc), is that players are paid for their efforts. In professional sports (for example in the NFL), the main goal is to win a championship. In other leagues (non-professional), goals may vary from learning leadership skills, sportsmanship, or working well with a team. In professional sports, players are paid, not to be an idol to the public, nor are they being paid to be a "nice" person. They are being paid to help the team win, and that is the ultimate achievement... And the rest just comes with the job.