I think when dealing with situations where people make bad decisions, it is important for everyone, whether parents, bosses, corporations, or governments, to understand exactly what they are trying to accomplish in instilling discipline.
It's very easy in a situation like this to come out with the attitude of, "Player X did something horrible, he should be punished for it." We see this fairly often in regards to situations where public figures makes mistakes. There tends to be a large outcry for immediate action to simply "get rid of the problem". This is the response that you typically see on twitter, facebook, and the like.
But let's look at what the point of instilling discipline is. It's not to simply punish someone and take away what they have earned or positions they have achieved. The real purpose of discipline is to allow the person to learn from the behavior that they previously exhibited, understand where they made their errors, and hopefully be able to proceed in the future without doing the same things again. The end goal is to guide the person towards being someone who is able to produce more positive results for society, instead of negative results.
An important consideration to make with regards to this is that simply labeling someone in a situation like this tends to create a number of problems that prevent this from happening. In both the Peterson and Rice situations, it's very easy to jump to calling either one of them a "monster", a "disgrace", or any number of different terms that have probably been thrown around. But this doesn't actually create any positive momentum to create a better result from it. Rather, the proper understanding is that these are people who made very bad decisions at one or more times in their lives, and we need to understand why that happened in order to change it. Simply labeling the person makes those flaws into an inherent flaw in the person, and prevents us from working with them to resolve the situation in the future.
So in short, what I'm getting at is that we need to look at the purpose of why we discipline people. AP needs to look at the purpose of why he disciplines his son. Goodell needs to look at the purpose of why he disciplines his players. Simply wanting someone gone because of mistakes they've made doesn't actually help to resolve the situation. Discipline has to be handled on a case by case basis, because the nature of what is called for will vary based on the specifics of the case.
I'm not saying that Peterson should escape punishment in this situation or that the NFL should not have a disciplinary policy for these things. But I think that the NFL, along with a number of other institutions, need to look at the reasons for discipline, as opposed to simply using the "label and remove" method that I described above.