Some people would prefer a theater of acting sad, holding your hand and pretending to be an old friend in a strange environment, other would prefer to receive the hit at their home and cry in private. Each people is different.
"First, do no harm."
Physicians, at least in Western medicine, are bound to the principle of non-maleficence. Their obligation is to take the approach (by action or inaction) that does the least harm to most people, NOT what helps the most people. This is an important, but subtle distinction.
This principle prevents, e.g. administration of a treatment that would cure 51 of 100 patients of a chronic but not debilitating condition while killing the other 49.
It's hard to imagine the quantifiable positive ill of telling someone something in person that they'd rather receive in email. But the underlying principle here is not in helping as many as possible but, overall, hurting as few.
After begin attacked by an enraged patient (or their relatives) I can understand how the physician could develop a phobia to put him/herself in a similar situation again. Just being forced to return to the same workplace day after day can be emotionally draining and very difficult for the physician.
The biggest problem of using e-mail in this case is keeping the privacy of the patient.
This was just them pulling a stupid, and the hospital apologizing for it.
Not only is email not able to be kept private, you also can't guarantee the patient is going to get it in any sort of timely manner, or in fact, at all. It's also ensuring that no follow up questions can be asked in a timely manner.
It'd be prime material for a Tom Lehrer song involving lawyers, malpractice law, ISPs, IT professionals, email protocols, and similar related ilk. Just this simple "proposal" is blowing my mind.
Subject: You're dying. Message: 2-3 weeks. Cancer. Please see attached invoice.