Is this a correct usage of a semicolon?

Wrong.

It's acceptable, but there's no need for any punctuation.

No, it's not. What follows a semicolon has to be able to stand as a sentence. A semicolon separates two main clauses that are interdependent. Since they are both main clauses, each has to be able to be read as a complete sentence. What follows your semicolon isn't a main clause and can't be read as a complete sentence because there's no verb.

Here's an example of correct usage of a semicolon:

* I'm not just incredibly idealistic; I'm smart, too.

The "not just" part of the first main clause makes it lean on the second main clause, and the "too" part of the second main clause makes it lean on the first main clause. That makes them interdependent clauses. Each is a complete sentence, but saying either one by itself sounds like something's missing. Now, had I used a coordinating conjunction like "but" between them, I could have used a comma, but without a coordinating conjunction, putting a comma there would be a comma splice, so I have to use a semicolon.

No, you would use a comma.