Based on the judgment on Case No. xxxx (X v. Y), would the independence of such authorities be called into question?
There are two main components to the sentence:
- the judgment on Case No. xxxx (X v. Y)
- the independence of such authorities
The sentence structure looks like this:
Based on X, would Y be called into question?
Rephrasing it to use the other phrase results in this revised structure:
Would X cast doubt on Y?
Therefore, what you want is this:
Would the judgment on Case No. xxxx (X v. Y) cast doubt on the independence of such authorities?
Note that, in the revised structure, you can simply replace cast doubt on with raise doubt over.