"To kill a fly with a..."?

... tactical nuclear weapon?

Google autocomplete suggests "sledgehammer", "bazooka", and "cannon". I think "sledgehammer" is the only one I've actually heard used, and I suspect it's what you're looking for.

A Korean proverb exists which states...

Do not draw your sword to kill a fly.

Actually the idiom goes, "to break a (butter)fly on the wheel". The wheel in question being a torture device, for humans rather than flies. From Wikipedia:

The breaking wheel, also known as the Catherine wheel or simply the wheel, was a torture device used for capital punishment in the Middle Ages and early modern times for public execution by bludgeoning to death.

The sledgehammer mentioned in the top answer is from a different idiom entirely, "cracking a nut with a sledgehammer". Same idea, of course, but the idioms are not to be mixed and matched at will except for comedic effect.

In German and Russian, there's also "to shoot sparrows with cannons" (mit Kanonen auf Spatzen schießen/стрелять из пушки по воробьям). That one hasn't caught on in English, though.

It could be "kill a fly with a sledgehammer", "kill a fly with a cannon", "kill a fly with a bazooka".

To reply to the additional question, "to kill a fly with […]" is the usual phrase. There are other phrases that are used to mean "using something that is excessive".

Use a surgical knife to slice bread.

One that I heard once and I liked a lot was "to swat a fly with a Buick".