Now, think of this situation in terms of your own work. I think that is the best way to consider it, since at bottom, it really is a Human Resources/work/management issue. How much joking at the expense of management is going on? How much open disagreement? How much of either would you tolerate if it was you in charge? If you didn't get rid of someone, what would you do instead? Those are the considerations that President Obama had to undertake.
Now, of course it wasn't that simple. Other people reported, then offered opinions and advice. What was written by the Rolling Stone reporter was quoted to support one view or another, often out of context. What McChrystal said was often confabulated with statements made by his staff members. The whole thing rapidly became bigger then it really was. That is why you have to read the original article--to see the quotes and issues in context.
Now, Mr/Ms. CEO, what do you do with this really smart employee, this manager with documented successes, but an occasionally difficult personality and with more difficult personalities lurking on his staff, who has some differences with others in the company about strategies? That's the question that you have to ask. How much of this sort of orneriness do you want and/or tolerate among your people?
Here's my answer: If this had stayed in-house or if publicized, had not blown up way bigger than actual size, I might just ask the general to keep a better track on his staff, keep his own doubts private in the chain of communication, and ask what I could do better to help him in his work. With it blowing up the way it did, even though at bottom it was not that big a deal, it became a big deal. When it became this big deal, it began to create uncertainties about the top leadership. In that case, which is what happened in reality, I would reluctantly accept his resignation. When I did, I would take time to talk about what a great leader he was and how much success he created--that he has been and hopefully will be continue to be an asset to the country.
In all of this, we missed something of an opportunity--the opportunity to revisit our entire strategy in Afghanistan and consider whether or not we should be there at all. This could still happen as a change in leadership is always an opportunity for assessment.
PS: I did not forget about the Constitutional issues. The founding fathers placed civilian control over the military, for better or for worse. This is the frame we work under, and this is the frame I answered the question in. The President is the Commander and Chief. He is the CEO. The military is a division that while working quite independently at times, still in the end, reports to the POTUS. Together they work on plans and strategy and work out their differences. Together.