Was the Galactic Empire a Sith Empire?

It is quite ambiguous even in-universe, but let's clear this up once and for all using a simple metric: Palpatine, the Emperor, was a Sith. Therefore, it was a Sith Empire.

But trying to parse whether it was a direct continuation of the Sith Empires of old… that's harder. If you want a TLDR before I dive into detail, the simple answer is that the Galactic Empire was a continuation of the ideals of the Sith Empires but as for being the same governmental body having changed forms over the millennia, the answer is no.

Legends and Canon

Since the question doesn’t specify canon or Legends, I'll do my best to tackle both. First, since there's a lot more established there, I'll begin with Legends. I'll also start with Legends since that question is a bit easier to answer with confidence.

Legends

The Original Sith Empire in Legends was largely destroyed in the Great Hyperspace War, except for a group of them led by Darth Vitiate which survived and rebuilt to become the Sith Empire encountered during the Old Republic MMO. While others had built Sith Empires—such as Exar Kun and Darth Revan—they were largely pretenders. They had nothing really in common with the True Sith other than they called themselves Sith.

The first Sith Empire rose, then Vitiate kept it going under an extreme restructuring. It was still the same Empire, but Vitiate's Sith Empire was dissolved/defeated under unknown circumstances two-thousand years prior to A New Hope. After that, a Sith Lord named Darth Ruin, previously a Jedi Knight, created his own Sith Empire. These New Sith might be a continuation in that they share the same basic commitment to the Sith Tenets as Vitiate's Sith, but as for being a direct successor as in a member of the previous Empire, no, it was not. Darth Ruin only uncovered a holocron's teachings. His Sith held to the same ideals, but by virtue of being a wholly new Empire, it was different.

Then we enter Darth Bane. The Sith ripped themselves as an Empire apart, and Bane was the sole survivor. But instead of building another Empire, he went into hiding and laid the foundation for the Rule of Two, which Sidious/Palpatine belonged to. Since the Sith went for so long without an Empire, when Sidious restructured the Republic into the Galactic Empire, it wasn't even a continuation of Darth Ruin’s Sith.

Canon

Now, Canon. This is why I saved it for second because it’s not nearly as fun a story to write all out. There were multiple Sith Empires, such as Exar Kun’s and the Brotherhood of Darkness, but the circumstances under which they were formed are still not known. The answer to the question, however, is the same as it was in Legends: The Galactic Empire is not a direct continuation but upholds the same ideals. Unless of course, you count two people with lots of money but no subjects, no army, and no real domination over the galaxy to be an ongoing empire, but the answer to that is probably no as well.

Ultimately, while the Galactic Empire was a Sith Empire, it was all Palpatine's. Built from the corpse of the Republic, as well—the final insult to one of the Sith's ancient enemies. If anything, it's more of a continuation of the Republic, since the same government was in power, just reshuffled severely.

In conclusion, while the ideals of the Sith Empires were continued in the Galactic Empire, the governmental structure and the Empire itself were distinct and new, shaped primarily by Palpatine's vision and ambition.