Yes, he was likely condeming the Witch-King to the Void. Most of the villains in the Lord of the Rings end up going there. As you say, the Silmarillion confirms that Sauron ended up in the Void:
"Sauron had a part, and was only less evil than his master in that for long he served another and not himself. But in after years he rose like a shadow of Morgoth and a ghost of his malice, and walked behind him on the same ruinous path down into the Void.”
This probably did occur with the intervention of the Valar:
The realm of Sauron is ended!’ said Gandalf. The Ring-bearer has fulfilled his Quest.’ And as the Captains gazed south to the Land of Mordor, it seemed to them that, black against the pall of cloud, there rose a huge shape of shadow, impenetrable, lightning-crowned, filling all the sky. Enormous it reared above the world, and stretched out towards them a vast threatening hand, terrible but impotent: for even as it leaned over them, a great wind took it, and it was all blown away, and passed; and then a hush fell.
(Return of the King
Our hint is the "great wind" which is clearly meant to provide an association with Manwe.
Something very similar happened to Saruman:
To the dismay of those that stood by, about the body of Saruman a grey mist gathered, and rising slowly to a great height like smoke from a fire, as a pale shrouded figure it loomed over the Hill. For a moment it wavered, looking to the West; but out of the West came a cold wind, and it bent away, and with a sigh dissolved into nothing.
(Return of the King)
Tom Bombadil sent the Barrow-wight to the Void:
Tom stooped, removed his hat, and came into the dark chamber, singing:
Get out, you old Wight! Vanish in the sunlight!
Shrivel like the cold mist, like the winds go wailing,
Out into the barren lands far beyond the mountains!
Come never here again! Leave your barrow empty!
Lost and forgotten be, darker than the darkness,
Where gates stand for ever shut, till the world is mended.
At these words there was a cry and part of the inner end of the chamber fell in with a crash. Then there was a long trailing shriek, fading away into an unguessable distance; and after that silence.
(Fellowship of the Ring)
But this shouldn't surprise us. Most likely, all Men go to the Void. Even Gandalf did, in mortal form.
Then darkness took me, and I strayed out of thought and time, and I wandered far on roads that I will not tell.
(The Two Towers)
The Gift of Men is precisely that they are not bound to the world.
It is one with this gift of freedom that the children of Men dwell only a short space in the world alive, and are not bound to it, and depart soon whither the Elves know not
That said, there are other things in the Void besides just Void: the Halls of Illuvatar are not a bad place.