I want to ask my girlfriend to pay the vet bill and replace her own belongings after she almost killed my dog
Why would you ask her to pay the vet bill?
Because I want to shame her in front of others for making my dog sick.
It's understandable that you would be upset and want your girlfriend to cover the vet bill when she is one responsible for making your dog sick and almost dying. It's expected that if you cause injury to someone or something you pay for the treatment necessary to make it healthy and whole again. While it's okay for you to request your girlfriend pay the vet bill, it isn't okay if you berate her in front of others for her mistake as that's being petty and vengeful and isn't appropriate behavior.
Well my girlfriend should've paid better attention and took care of my dog.
You're right, she should have been more conscientious with your dog and it is reasonable to expect her to pay for your dog's treatment. Anyone who injures a pet should help the owner pay for any bills, as it's the responsible and kind thing to do.