An apology is an expression of regret.
And any old regret will do? How about regret for the action? They might as well express regret that there wasn't enough mayonnaise on the sandwich they had for lunch.
It's fair to regret a certain consequence of an action, but not the action itself. When you say something that people decide (irrationally, in this case) is insensitive or offensive, there's nothing you can do subsequently that will appease them. The people who have chosen to be offended by the original tweet will likely remain steadfast in their choice. Even a "real apology" according to their bizarre concept of "apology" wouldn't appease them.
Remember, people choose for themselves what they find offensive. If you do something with good intentions, and someone chooses to be offended, it's reasonable to regret their response to your action, but there's no reason to regret the action itself.
Anyway, as soon as you challenge the "realness" or validity of a specific apology, it brings up the issue of what the purpose of an apology really is, especially an impersonal apology from a major corporation. To me, an apology is not necessarily an admission of guilt. An apology is an expression of regret.
What would you propose to be a "real apology" in this case? Do you want Bing to tweet out that they purposefully did this to slight the victims of the tragic earthquake and get retweets? Do you think that would be honest?
edit: typo