Assuming that the manager values all of these employees and wishes to retain them, he or she can use emotional intelligence as one tool in the management toolbox. Emotional intelligence, the topic of much of Daniel Goleman's writing, is a term that seems to be tossed about a great deal but disregarded in action, much like critical thinking. It is important, and it needs to be used far more than it is now. Emotional intelligence, to me, is having empathy or insight into the behavior of others and oneself and acting in a way that defuses harm and produces positive outcomes.
The manager who has been approached by the shaken employee needs to first have some empathy for that employee, understanding that she has been verbally attacked and threatened, and that she needs some assurance about the safety of the workplace, both mentally and physically, for herself and others. If this attack on her has just happened, returning her immediately to the place where the problematic worker is without having done anything about him yet is not going to be particularly reassuring. I would say that some discussion about getting this fellow some help for his problems (and meaning it) will be helpful, and I would be inclined to tell her to take a long lunch or a long break while you sort this out with the fellow. I also think that taking the "He will be punished" tone with her is a mistake, escalating the problem, particularly since she has already explained that he seems to be under stress. She is not approaching the manager in a vindictive mood, I would say, just an upset and frightened one. She needs to be calmed down, removed from the situation, and assured that the manager is on the case. We have all ready too many tales of employees going "postal," and that is likely to be her biggest fear, so words of assurance are the most important.
The manager needs empathy to deal with the troubled employee, too. Given that this troubled employee's problematic behavior is recent and reportedly caused by stress, something is clearly wrong. Approaching him in a spirit of helpful inquiry is the key to learning what the problem is and then helping him, if possible, to solve it. This should be done immediately after meeting with the reporting employee, not delayed for any other reason. A meeting should be private, not a public questioning or chastising. The manager must find out what the problem is and offer assistance. Many organizations have an employee assistance plan (EAP) that provides counseling for employees. Some problems can be solved simply by working with the employee. He might have a new child at home and does not get enough sleep. Perhaps his hours could be adjusted to allow him to sleep a bit later in the morning. He may be having marital problems, and all the manager can do is to sympathize. But the employee who can articulate what the problem is is on the road to being helped.
Having said all of that, the manager also needs to explain to this fellow that his behavior is unacceptable in the workplace, no matter what his reasons are. His workmates are upset and afraid, with good reason. The manager can ask this employee to use a little empathy too, asking "How would you feel if your fellow employees spoke to you and threatened you like this?" I would not at this point discipline or terminate. Assuming that the manager has the discretion to decide upon this, he or she should give the person a chance to work on the underlying problem and his behavior. It should be made clear that another incident will not be tolerated, though. Followup with this employee is a must, to see if he is getting assistance, what kinds of arrangements can be made to accommodate him, and so on. The manager needs to keep a close eye on his behavior, too, but this could be done in a supportive way, by stopping by the department and asking everyone how things are going.
Should there be a meeting to discuss violence and bullying in the workplace? I am not convinced this would be an emotionally intelligent step right away. The rumor mill has already let everyone know what has happened, and a meeting might very well keep people gossiping. After some time has passed, a manager might invite employees to a meeting to discuss what kinds of policies would work well to ensure that the workplace is safe for all.
A manager who is calm and empathetic can deal with this problem in the workplace. Emotional intelligence is certainly applicable to this situation.