M. Koszowy and D. Walton argue that there are two types of authority, epistemic and deontic, that can become confused and lead to fallacious arguments from authority. The epistemic authority involves expert opinion. The deontic authority involves administrative authority. Fallacious situations arise when such authorities are not adequately questioned often because of a confusion of these two types of authorities.
To study these two types of authority they use argumentation schemes. As an example the argumentation scheme for epistemic authority is as follows: (page 4)
Major Premise: Source E is an expert in subject domain S containing proposition A.
Minor Premise: E asserts that statement A is true (false).
Conclusion: A is true (false).
One tentatively accepts the conclusion, but then critical questioning should begin. They list six such questions as initial examples: (page 4-5)
Expertise Question: How credible is E as an expert source?
Field Question: Is E an expert in the field F that A is in?
Opinion Question: What did E assert that implies A?
Trustworthiness Question: Is E personally reliable as a source?
Consistency Question: Is A consistent with what other experts assert?
Backup Evidence Question: Is E’s assertion based on evidence?
Once a critical question has been asked, the other side needs to make a response. These critical questions are similar to the questions that the OP lists for consideration:
"How can you trust the opinion on any matter of an anti-vaxxer?"
"This doctor prescribes homeopathy so why should you believe any other thing about healthcare he says?"
By themselves, these questions are not fallacious. This critical questioning is an important part of the argument. As the authors note, "it would be a great error to treat experts as infallible". (page 18)
The next step is for someone supporting the doctor to make a response. A fallacious situation could occur if someone felt too intimidated or confused to adequately question the authority.
Koszowy and Walton list five areas where fallacies may be recognized: (page 18)
Our theory is that recognizing the fallacy should be based on five underlying factors: (1) the ambiguity between the two types of authority, the epistemic and deontic one, (2) the transitioning in arguments from authority from the one type of authority to the other, a transition that might easily passed unobserved without an awareness of the subtle matters of argumentation in it, as shown in section 6, (3) the confusion that might easily arise between the two types of authority in arguments in which such a transition occurs, (4) the unthinking and unquestioned transference of the deference properly attributable to the deontic authority, leading to acceptance of the epistemic authority, and (5) the overlooking of critical questions which may need to be
considered in response to the argument from epistemic authority.
From just these two questions it is unclear if any of these factors have occurred. That questions were asked at all is evidence that the argument did not involve a fallacy, but these questions may not have been the best questions to ask.
M. Koszowy and D. Walton Epistemic and Deontic Authority in the Argumentum Ad Verecundiam. (2019). Pragmatics and Society, 10(2), 2019, 151-179 Retrieved on July 3, 2019 from Douglas Walton's site at https://www.dougwalton.ca/papers%20in%20pdf/19Deon.pdf