The Quest to answer the OP's question ...
... has, alas!, been diverted 'long way around the barn' by, I deem, agents of The Enemy.
Before I make any assay as to the "reincarnatability" of the Witch King, I will speak towards the other issue, the fate of the Rings of Power after the One is destroyed, because --logically-- restoration of the Witch King could be achieved only if his Ring retained its power.
At the climax of Book 6, Frodo, standing on the very brink of Doom, rejects his quest and claims the One Ring for himself. Sauron is fully-aware of this, and of Frodo, and of His own desperate peril, immediately, and all of the eight remaining Nazgul abandon the battle before the Black Gate, and rush madly south to Mount Doom. They never make it: when Gollum falls into the chasm, the One is destroyed, and all of the Nazgul, shrieking impotently, "disintegrate" and blow away with the wind. When the One was destroyed, so was Sauron, but also the Nazgul: it is clear that the Nine Rings lost their power with the destruction of the One; hints and suppositions aside, there is no reason to think that the same fate did not apply to all the Rings of Power.
Now, the OP question: if the Quest had failed, or if the Rings of Power retained their magic, could/would the Witch King somehow/somewhen return?
On the one hand, impossible to say, as in, the author never made any declaration on the issue. On the other hand, there is enough evidence in the texts to permit some reasonable supposition.
My own opinion: in time, the Witch King would "regenerate"; his "death" on the Fields of Pelennor did not (completely) break the magic spell that bound his hroa and fea (body and soul) to his Ring, merely destroyed that specific incarnation, which could be slowly restored/replenished (in much the same way that Sauron required c. 1000 years to recover from his defeat at the end of the Second Age). Sauron's assistance, or even existence, would not necessarily be required, it would suffice only that the Witch King's own Ring (still) "worked".
A point of trivia of which some readers might be unaware: neither Merry nor Eowyn found "a pretty gold ring" on the ground after the Witch King got his block knocked off and went "poof" on the Fields of Pelennor; Sauron had dispossessed the Nazgul of their Rings: the Nine were in His keeping.
EDITED for spelling mistake, above: wich king is the Witch King ...
I'll offer some clarifications that might help.
This is a bit of a "grab-bag", and cut-short for clarity:
The barrow-blades are described as being, "wound with spells for the destruction of Angmar", or words to that effect: if this description does not appear in Book 5, then it's in Book 1, either after Bombadil first rescues the hobbits from the barrow, or in the wilds with Aragorn when he gets a peek at their equipment. I'm also sure that after Merry's blade breaks a passage reads something like, "happy would be he, who first wrought [Merry's barrow-blade] to know [that it would strike the blow that ham-strung the Witch-King]".
I pointed out earlier that the Nazgul are Nazgul because of their own Rings, not because of any "empowerment" by Sauron; please refer to my most-recent post.
Same post, I reminded everyone that the (remaining) Nazgul were destroyed when the One Ring was destroyed, not because destruction of the One destroyed Sauron, but because the Nine Rings lost their Powers when the One was destroyed. The eruption of Orodruin that ensued had nothing to do with the destruction of the Nazgul: they "disintegrated" in mid-air and, presumably, mid-shriek.
Gandalf's remarks ("the Nazgul stand or fall by their master") remain true: Sauron could only be overthrown by the destruction of the One Ring, which is the only event that could break the spell of the Nine Rings over the men who had become Nazgul.
Gandalf's words to Legolas in Rohan remain true: any Nazgul has become, unnaturally, not-mortal, so no "mortal wound" would appreciably vex one.
Glorfindel's Witch-King prophecy, "not by the hand of man will [the Witch King] fall" should not be misinterpreted: the prophecy is foresight, not a dissertation on the nature of the Nazgul.
The Nazgul hated and feared water, as well as fire: all natural forces accosted their unnatural existence; even more, they hated and feared a High Lord of the Eldar, fully revealed in his wrath and wielding fire.
When the Nazgul are overwhelmed in the flood at Rivendell, they are robbed of their shapes and forced to return to Mordor, rather than continue to pursue their errand: this can only mean that they were, if only temporarily, un-made by that torrent of elemental force; "shapeless", how did they cross Anduin, or any other intervening rivers? (rhetorical question).
Gandalf (and possibly others) comment on the "nothingness", or "emptiness", of the Nazgul: all of these remarks should be considered in the context of double-entendre; the Nazgul are devoid of (real) life, virtue and love, and free will; they are nothing but malicious automatons, abominate effigies enthralled to the Power of their Rings, and to the will of the Master thereof.
"All blades perish, which pierce that dreadful King", or words to that effect, is the reason given for the breaking-of-blades-against-the-Witch-King, specifically: this is merely an imprecation, the provenance of which remains unknown; it could have been bestowed by Sauron, or by the Witch-King's own sorcery, but it is strongly suggested that it is part-and-parcel with the unnatural nature not of Nazgul, which is dependent on the nature and Powers of the Nine Rings, but of the Witch-King-Nazgul-Chief himself (as if he were, to other Nazgul, what a superduperintendant would be to lowly superintendants).
"Subintendant" Morgul-knife-Frodo would be exactly like the Nazgul, only weaker: magically cursed into the form of the Ulairi, but not linked to any Ring of Power.
The Curse of the Dead of Dunharrow was empowered by two things: the "virtue" of the Stone of Erech, as a rescued hallow of Numenor, and Isildur's right to command that "virtue" (itself derived from two sources: Isildur was, by direct line of descent, the true heir of Elros, first King of Numenor; secondly, Isildur had personally rescued the Stone of Erech from the Downfall); in short, this curse is "The Divine-Right Stuff" (that no one at NASA can claim to have ...).
If I've left any loose ends, I'm sure I'll rue it.