Isn't grandmothering wonderful!
I always laugh and say it's worth all the struggles we go through raising our children just to get to be a grandparent! My spin on your story is that neither you nor your daughter did anything wrong. Dogs (especially small dogs) do occasionally bite young children, who have a way of pulling ears, moving too swiftly, or even poking at an eye. You were wise to pull your granddaughter away. On the other hand, it is easy to intensify fears in young children by an over-emotional reaction to a possibly risky or dangerous situation. You can understand that your daughter doesn't want her little girl to be afraid of everything.
I suggest the two of you sit and discuss the episode and other possible events of that nature. Ask your daughter if she thinks you made her fearful when she was a child, that you over-react, etc. And, if you think she is too casual about possibly dangerous experiences, let her know why you feel as you do. Chances are the two of you can reach the proper level of both protection and concern.
Our parenting advice is given as suggestions only. We recommend you also consult your healthcare provider, and urge you to contact them immediately if your question is urgent or about a medical condition.