I seriously doubt it, and I don't think any vet would. Unless there was negligence on the owner's part. So say the dog was often violent, and the owner knew this, but did not disclose it to the vet, or lied about it, then perhaps.
But in general, a vet would never do this. MANY pets will become fearful and aggressive at the vet, and it is the job of the practice to have a system in place to keep the vets from becoming harmed. So if the practice slipped up and allowed a pet to cause a vet damage, that is more the practices fault than anyone else, as keeping this from happening is a big part of a practice's job. And why would a vet want to harm a pet or owner just because the pet acted like many pets do, and an accident happened? Most vets love animals, and would not want to blame them for doing something many animals would do if given the chance, simply out of fear.
Vet practices also have insurance that would cover any medical costs and missed wages, so there wouldn't be much to sue an owner for. And if the vet decided to sue anyone, they would likely sue the practice, who has insurance to cover being sued, and they would get more money, and have a much better chance of winning.
Also, that vet's reputation and career would be ruined if they decided to try and sue an owner for a bit, or if they tried to get a pet put down. I doubt any practice would want them, because no one would want to go to a vet who might try to sue them or get their animal put down if the pet was aggressive. People trust their vet to be able to handle their animal, even if the animal is aggressive and fearful.
It just wouldn't make sense for a vet to do that, unless the vet was really upset with the owner for some sort of negligence, such as purposefully lying about bite history, and knowingly endangering the vet or something. Or maybe if you brought your aggressive dog in and let them go in the lobby to just attack everyone or something.
I work at a vet's office, and vets absolutely have been bitten, but it's really hard to even imagine a situation where any of our vets would decide to sue or try to get an animal destroyed. I just don't see that ever happening except in the most extreme circumstances of an owner committing extreme negligence, and likely not even then.