Learn Russian: Судья настоял на новой проверке.

The verb настоять (to insist) requires the preposition на followed by the Prepositional case to indicate what is being insisted upon. This is a set rule for verbs of insistence and conviction. Even though на can sometimes take the Accusative to show direction, here it is locked into the Prepositional.

Since судья ends in -я, why does the verb настоял take a masculine ending instead of the feminine настояла?

Судья is grammatically a masculine noun because it originally referred to a male role, even though it declines like a feminine noun ending in -я. Modifying adjectives and past-tense verbs must agree with this masculine gender. If the judge were explicitly known to be a woman, you could use настояла in modern Russian, but masculine is the standard default.

Why use the perfective verb настоял instead of the imperfective настаивал?

Using the perfective настоял shows a completed, successful action: the judge put his foot down and his demand for a new inspection was finalized. If the sentence used the imperfective настаивал, it would mean he was insisting over a period of time, but we wouldn't know if his demand was actually met.

Does the word проверке always mean an inspection?