No, but contrary to Robert's answer I think it would be a sign of good leadership to take sides (and bring employees and the community together). By taking sides I don't necessarily mean taking one side on all issues. It's hard to go into specifics without knowing the details of all issues. I think it's reasonable for issues that affect SE to the extent that moderators resign en masse or where users feel excluded for the CEO to get involved, for example by leading a review.
On the particular issue where a moderator was fired with a lot of accusations in public, it may be good to step in. By stepping in I mean reviewing the steps that were taken and seeing if something went wrong (in the entire process starting with the firing up to how it is being handled now). The outcome of that review can be used to see if internal policy needs to be changed and could be used to ease tensions in the community.
While this does legitimately involve real people, Stack Exchange is not a government organisation that handles covert operations. Indeed, it is a private company and they don't have to communicate anything if they don't want to, but they can do so in their own interest. Without jeopardizing the privacy of any employee they can make anonymised statements on their internal process. While that may not satisfy all, I think it's a good start in coming together.
Rather than arguing the point on leadership myself, let me quote from Scott Gregory's The Most Common Type of Incompetent Leader in Harvard's Business Review:
Researchers have studied managerial derailment — or the dark side of leadership — for many years. The key derailment characteristics of bad managers are well documented and fall into three broad behavioral categories: (1) “moving away behaviors,” which create distance from others through hyper-emotionality, diminished communication, and skepticism that erodes trust;
[...]
However, my friend was describing something arguably worse than an incompetent boss. His manager was not overtly misbehaving, nor was he a ranting, narcissistic sociopath. Rather, his boss was a leader in title only — his role was leadership, but he provided none. My friend was experiencing absentee leadership, and unfortunately, he is not alone. Absentee leadership rarely comes up in today’s leadership or business literature, but research shows that it is the most common form of incompetent leadership.
And more to the point, though the entire article is worth reading in my opinion:
Like the provost in this example, many organizations don’t confront absentee leaders because they have other managers whose behavior is more overtly destructive. Because absentee leaders don’t actively make trouble, their negative impact on organizations can be difficult to detect, and when it is detected, it often is considered a low-priority problem. Thus, absentee leaders are often silent organization killers. Left unchecked, absentee leaders clog an organization’s succession arteries, blocking potentially more effective people from moving into important roles while adding little to productivity. Absentee leaders rarely engage in unforgivable bouts of bad behavior, and are rarely the subject of ethics investigations resulting from employee hotline calls. As a result, their negative effect on organizations accumulates over time, largely unchecked.
