Yes, she should have, anyone who doesn't realise this is a racist term obviously must still be living pre 1960s.Or has clearly never experienced racism in their life. The term is offensive! What's more shocking is not that she said it was that she didnt want to apologise!!
Private or not it shows her way of thinking. Made in a joke or talking about someone's hair, the term is still offensive. She obviously has very little emotional intellignce for the rest of the nation.
We should not be condoning racist terms, as a influential powerful broadcaster, the BBC need to help lead the way by showing others that this is unacceptable and I think getting rid of her is definitely a good step. If the BBC had not taken action then they would continue to be seen by some, as out of touch.