This is a really nasty thread. Oh, the irony of women sending another woman waspish messages about the need to be kind.
OP, you know your daughter really well. I'm sure she is the kind, sensitive girl that you say that she is. However, even the kindest and most sensitive person sometimes makes mistakes. We all interact with people occasionally in ways that are suboptimal, and sometimes even adults misjudge tone or situation. Such incidents are not 'bullying' - because bullying implies a sustained campaign of some sort. They can, however, be nasty, hurtful mistakes.
I think the way to handle this is to take it seriously as an out of character mistake. Make sure that your DD doesn't minimise the upset that she has caused, and get her to write a note of apology to the girl. This will hopefully ensure that this one-off doesn't get repeated and that she is very careful about what she sends in future. Then, for God's sake, let it go. This error doesn't define your daughter, and nor does the casually applied label of 'bully'. Continue to promote and foster her kind, generous self and no-one will take an accusation like that seriously. Maybe also encourage her to develop other friendships than that with this girl.
And now you: you sound like you are excessively concerned about this allegation, and your reaction seems disproportionate to the situation. If you are grieving, this is understandable. But please, try not to worry so much what others think. 