Not every failure needs a post-mortem. What do you want him to say: "We've determined that all employees -- including me -- need to attend sensitivity training to ensure that this does not happen again"? Or perhaps: "We will install filtering software to catch this kind of message before it ever reaches our customers in the future"?

No - it's not an operational failure to be picked at until a systemic solution is arrived at, it's a human failure. His failure specifically, and he acknowledges this.

He's human. Customers are frustrating as all get-out; you want to do the very best by them; and your best is NEVER good enough. Chances of this happening anywhere is non-zero, and there is no indication this incident has a frequency, so you can't really reduce it. Not to mention, it's pretty probable that he has learned his lesson and thus decreased the frequency of such incidents, but we won't know until, at least, the third time.

That's important when you're dealing with a systemic problem that needs some specific effort to reform or eliminate - which is not the case here. It's a solid and sincere apology, offered without reservation or blame for others, and clearly heartfelt. Most people seeing something like that are satisfied. Turning something decent into something obviously and needlessly formulaic actually undermines its quality.