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.