BGIF-
The NFL absolutely has the right to prevent him from working. The CPA and the doctor still might have the ability to work, but the public can pick and choose whether or not they want to do business with someone who has been convicted of a crime. Goodell is looking out for the league as a whole because a negative public view can adversely affect the whole league.
If that same CPA you used as an example works for a large firm like HR Block, his job is most certainly in jeopardy due to a DUI conviction. If the doctor practices at Kaiser Permanente, I wouldn't be surprised to find that a felony conviction would also prove costly.
The NFL is working in the same capacity. Additionally, Goodell is well aware of the statistics of recidivism and I am sure he wants to make sure old habits do in fact die, so as not to have another incident.
Yes, he served his time, but anyone who has applied for a job knows, that question is asked on every application you will ever fill out. "Have you ever been convicted of a felony?" And that is a question that can (and should) stick with someone where ever they go. It has nothing to do with violation of rights - it is purely business.
Goodell doesn't want to see Animal Rights protests outside of every stadium Vick plays in or widespread boycott of the team he is playing for. It will happen. PETA, the ASPCA, and the Humane Society of America are all very powerful lobby groups and they can create a lot of problems for the NFL. This issue was a very hot blooded and polarized topic and Goodell is well aware it won't go away quietly.