Is the use of a semicolon in the sentence 'The south sided with the King; the north with the usurper.' correct?

I would use a comma, not a semicolon in the example you are providing. As user Edwin Ashworth was suggesting in his comment, often it is intuitively correct to use a comma instead of a semicolon. The strict use of semicolons in such cases seems outdated to me today.

If a comma were used instead of the semicolon, your sentence would arguably contain a 'comma splice' (though gapped structures, as this is, are rather different). Some traditionalists might say that that is unacceptable. However, many (I'd say more enlightened) Anglophones are prepared to accept comma splices where the sentence is reasonably short and expresses closely related truths (eg Man proposes, God disposes). This is particularly so when the second clause is contrastive (Wallraff, in the book mentioned below, suggests that a comma can be used to join independent clauses when “the whole point of two clauses is to contrast negative and affirmative assertions”.

I'd say that is true here. Barbara Wallraff in her book "Word Court" goes further than 'a comma may on occasion be acceptable', commenting on the sentence

It's not a comet, it's a meteor.:

[P]unctuating this sentence with a semicolon would be like using a C-clamp to hold a sandwich together.

(On the other hand,

Holmes found Moran, he was dead.

is best avoided.)

But using a semicolon in your example is probably even less criminal than using a C-clamp on a sandwich.

Your sentence is both elegant and totally correct! It is not mandatory that the semicolon should be followed by a complete sentence, especially when the verb before the semicolon is repeated after it.

Example:

He was a war-monger; his son, a pacifist. (He was a war-monger; his son was a pacifist)

Moreover, the semicolon serves to suggest a pause of half a beat for the reader, and can be followed by 'and' or 'but' where appropriate, thereby not forming a complete sentence in the second part, as in "the south sided with the king; and the north, with the usurper." The use of 'and' as well as the comma after 'north' is optional.

It is often the writer's choice whether to use the comma or the semicolon, or use neither.

Example: 3 correct ways to write the same sentence --

He read a lot and gained a lot from the reading.

He read a lot, and gained a lot from the reading.

He read a lot; and he gained a lot from the reading.

(See how he is added to balance the pause caused by the semicolon)

So you see how (once your basic grammar becomes strong and you become confident of avoiding errors) it's all a matter of style!

The convention is that only clauses which are independent can be separated by a semicolon. Your second clause is not independent and therefore should be separated from the first with a comma.

Your second clause is elliptical. The antecedent of an elided element (here it's the verb "sided") must appear in the same context; a semicolon would mark the start of a new context.