I do not think there is a definite answer, only speculation.
While other answers may be right in that perhaps Force Ghosts may not be a concern to Vader, despite that we do eventually learn they can evolve to be able to interact in the material world (Yoda: Force Lightening, The Last Jedi), that doesn't mean that there are not other kinds of "entities" or things that exist beyond "life" that could of once intimidated him.
In fact the Marvel Comic series Darth Vader: Dark Lord of the Sith (2017) takes place right after the events in Revenge of the Sith where Darth Sidious gives a broken Anakin Skywalker an ultimatum: continue his transition into Darth Vader or die!
Vader chooses life and is then set up on a journey that only painfully continues to allow him to transition into the man that we come to know at the beginning of A New Hope.
After hunting down Jedi, bleeding that Jedi's lightsaber's kyber crystal to make it his own, he then ends up on Mustafar again where he comes across a helmet.
It once belonged to a legendary Sith (or artist), Darth Momin, but somehow Momin's consciousness is attached to the helmet and Momin tells Vader his story and how & why he should build a Castle with a Force-Sensitive tuning fork here, so that he can find and enter a portal into the spirit world and find Padme` again...
Vader does everything Momin asks and is strong enough to open the portal, but Momin comes back to life and Vader is forced to fight and kill Momin or risk loosing Padme again. Vader of course wins, but as he travels through the spirit world that also depicts echoes and fragments of memory from the past, he does finally find Padme`, but she rejects him and escapes his reach!
It appears that after this point, Anakin accepted his role as Vader, but he was also contested again and again through Sidious...
In another Marvel Darth Vader Comic (2015), which takes place in between A New Hope & The Empire Strikes Back, Darth Sidious is furious with the destruction of the first death star and so he doesn't think his apprentice is worthy.
He somehow knows or comes to meet a Dr.Cylo, who claims he has created cybernetic beings who can mimic the Force well enough, claiming that one doesn't need the force to beat it! When Vader comes to fight these cybernetic beings, he proves they are no match for the real Force, but he unsuccessfully kills Dr. Cylo, as also being cybernetic, he is able to transfer his conscious to another body! But later, Vader does find Cylo and destroy all of the clone bodies!
There are also things introduced in the Star Wars universe such as Dark Side Wraiths, which are shadowy entities of the souls or essence of dark side people/energy somehow left behind.
The Force Awakens *ALMOST did something similar with Anakin Skywalker, as this concept art depicts:
Death is a Powerful Motivator
But the reason I brought these instances up, is because Anakin's turn greatly relied on the belief in life extension, as once presented by Darth Sidious in Revenge of the Sith.
PALPATINE: Did you ever hear the tragedy of Darth Plagueis the Wise?
PALPATINE: I thought not. It's not a story the Jedi would tell you.
It's a Sith legend. Darth Plagueis was a Dark Lord of the Sith, so
powerful and so wise he could use the Force to influence the
midi-chlorians to create life. ... He had such a knowledge of the dark
side that he could even keep the ones he cared about from dying.
ANAKIN: He could actually save people from death?
PALPATINE: The dark side of the Force is a pathway to many abilities
some consider to be unnatural.
Darth Sidious whole gamete has been about immortality, as he's constantly looking for relics, artifacts, or other sources of dark energy force power to immortalize himself and in which he uses to motivate Vader along, as he is also desperate to save or find the one's he lost...
And so the line might be at least curious to Vader, in that Vader himself does know the power of striking one down (Sidious pit Vader against Count Dukoo in Revenge of the Sith to continue to push Anakin towards the Dark Side, as striking someone down in anger and hate is way to do that and because Obi-Wan nearly killed Anakin in at the end of Revenge of the Sith), but also because Vader was aware and was still searching with Sidious for something "beyond" corporeal reality, as depicted in The Aftermath Trilogy.
It's just Vader's search for what he lost, was of the dark side way, and therefor it was considered by the light side as "unnatural" (Force philosophy is debatable IMO and apart of Star Wars story progression) and he realized that instead of continuing to looking elsewhere, all he had to do was save his son & daughter from the same painfully suffering fate that Sidious had done to him.
But beyond Vader contemplating his own story through the death of Obi-Wan, we really have no way of knowing what was going through his mind or how he took the line himself.