Let's look at the original sentence that led to this query:

Approved April 24, 1800, the act of Congress that made provision for the removal of the government of the United States to the new federal city, Washington, D.C., also established the Library of Congress.

Let's break this sentence down into its pieces. If the piece is just a modifier, then it is not part of the sentence's core structure. Since the poster's question is one about sentence structure, let's eliminate those pieces, and see what is left.

(Approved April 24, 1800,) -- modifier describing "the act."

(the act) -- noun...possibly the subject of the sentence. Hold off on eliminating it.

(of Congress) -- prepositional phrase. Cannot be the subject.

(that made provision for the removal of the government of the United States to the new federal city, Washington, D.C.,) -- big modifier describing "which act of Congress it was."

(also) -- modifier describing "established"

(established) -- verb...could be the main verb. Hold off eliminating it.

(the Library of Congress.) -- what the act established...could be the object. Hold off eliminating it.

So, now, here is my sentence in core structure mode:

"The act established the Library."

Is this is a complete, grammatically correct sentence? Yes it is! You have properly eliminated the extras in the sentence, and boiled it down to a single independent clause.

So, in fact, there aren't two independent clauses in this sentence. It simply feels like there are, because it's so long. The GMAT does that ALL THE TIME!

Here's a simpler sentence with the same structure:

The dog that likes steak also likes ham.

This is the same general structure of the GMAT sentence at the top of this post, but because it is so short, it is much easier to see that it is one independent clause.

By the way, you NEVER want to have this:
Independent Clause, Independent clause.
If you want to separate two independent clauses, you need either:
A) comma + conjunction
B) semi-colon