We don't know; I'm inclined to argue "no", though my argument is less solid than I'd like.
The Morgul-spell
Unfortunately we know little about the enchantment placed on the Morgul-blades. What we do know is that it's tied very closely with the long-term effect of wearing a Ring of Power. The best description of this "wraithification" process comes from an early draft of what would become chapter 2 of Fellowship of the Ring:
Yes, if the Ring overcomes you, you yourself become permanently invisible - and it is a horrible cold feeling. Everything becomes very faint like grey ghost pictures against the black background in which you live; but you can smell more clearly than you can hear or see. You have no power however like a Ring of making other things invisible: you are a ringwraith; and your clothes are visible, unless the Lord lends you a ring. But you are under the command of the Lord of the Rings.
History of Middle-earth VI The Return of the Shadow Chapter III: "Of Gollum and the Ring"
This is a very early draft (1938ish), and the full text includes many details that were later changed, but the idea of a ringbearer slowly becoming wraithlike stayed in-place, at least with Men, so I'm inclined to take this description as it stands. In fact, it's worth comparing this to an observation of Frodo's, not long after he was wounded:
Frodo threw himself down, and lay on the ground shivering. His left arm was lifeless, and his side and shoulder felt as if icy claws were laid upon them. The trees and rocks about him seemed shadowy and dim.
Fellowship of the Ring Book I Chapter 12: "Flight to the Ford"
It would appear, then, that the effect of the morgul-wound is extremely similar to the "wraithification" effect of the ring1. In fact, in his early drafts, Tolkien suggests that the two effects (that of the wound and that of the Ring) feed off one another; he writes of a conversation between Gandalf and Bingo (proto-Frodo):
'What happened at the ford?' asked Bingo at last. 'It all seemed so dim somehow, and it still does.'
'Yes!' answered Gandalf. 'You were beginning to fade. They would have made a wraith of you before long - certainly if you had put on the Ring again.
History of Middle-earth VI The Return of the Shadow Chapter XII: "At Rivendell"
What about Dwarves?
As was already pointed out in the question, Dwarves appear immune to the wraithifying effects of the Rings of Power. This is another idea with a long history in the narrative; in an early draft, Tolkien writes:
The dwarves is is said had seven, but nothing could make them invisible. In them it only kindled to flames the fire of greed, and foundation of each of the seven hoards of the Dwarves of old was a golden ring.
History of Middle-earth VI The Return of the Shadow Chapter III: "Of Gollum and the Ring"
And in a later draft, he writes a slightly weaker version of the same:
The dwarves proved tough and intractable: for they do not lightly endure any obedience or domination (even of their own kind). Nor are they easily made into shadows.
History of Middle-earth VI The Return of the Shadow Chapter XV: "Ancient History"
Although I'm clearly engaging in some pretty heavy speculation here, it would seem likely that the Dwarves would, indeed, not be susceptible to that effect of the Morgul-blade2.
1 This may sound like me speculating, but I have Christopher Tolkien on my side; in his commentary some early drafts, he says:
[T]he idea has now entered that the wound of the Ring-wraith's knife produces, or begins to produce, a similar effect to that brought about by putting on the Ring: the world becomes shadowy and dim to Bingo [later Frodo], and at the end of the chapter he can see the Riders plain, beneath the black wrappings that to others cloak their invisibility.
History of Middle-earth VI The Return of the Shadow Chapter XI: "From Weathertop to the Ford"
2 Although, in fairness, it's not entirely clear who is susceptible. The spell certainly doesn't always apply, as Micah points out in an answer to a related question, but without a larger sample size we can't say much conclusive on the matter.