The Nerevarine fulfilled the actual contents of the prophecy, namely destroying the Tribunal. However, popular opinion was that the Nerevarine would accomplish things beyond what the prophecy actually said, like driving out the Imperials, and these were not accomplished.
The aftermath involving the destruction of Morrowind was not part of the prophecy, but was a consequence of dethroning Vivec.
Figuring out if the Nerevarine fulfilled the prophecy for him is an interesting question because the existing prophecies, and what people say the prophecies mean, are sometimes wildly divergent.
The most common version of the prophecy is very vague
The main Nerevarine prophecy (also called "The Stranger") is much more vague. This is how it appears in the decoded package attachment given to Caius Cosades (reformatted for clarity):
The Nerevarine Prophecy:
This most common verse is one collected from Ashlander tribesmen. Our incomplete sources suggest that the Ashlanders have a strong tradition of dream-visions and prophecies, and that, among the Ashlanders, there are many other verse prophecies concerning the Nerevarine.
When earth is sundered, and skies choked black,
And sleepers serve the seven curses,
To the hearth there comes a stranger,
Journeyed far 'neath moon and star.
Though stark-born to sire uncertain,
His aspect marks his certain fate.
Wicked stalk him, righteous curse him.
Prophets speak, but all deny.
Many trials make manifest.
The stranger's fate, the curses' bane.
Many touchstones try the stranger.
Many fall, but one remains.
This is also detailed in Nerevarine cult notes
Note that this prophecy can be interpreted in a number of ways, especially phrases like "Many fall, but one remains".
The most common interpretation of the prophecy is "a local superstition", rather than a direct reading
The main part of the decoded message:
A local superstition holds that an orphan and outcast, a youth born on a certain day to uncertain parents, shall unite all the tribes of the Dunmer, drive out the invaders of Morrowind, and shall reestablish the ancient laws and customs of the Dark Elven nations. This orphan and outcast is called in legend the "Nerevarine," and is supposed to be a reincarnation of the long-dead Dunmer General and First Councilor, Lord Indoril Nerevar.
As we can see, local belief does not match the most commonly available version of the prophecy.
However, there are those who say their view on the prophecies are based on "lost prophecies" that have either been forgotten or deliberately suppressed by the Temple. From "Nerevarine cult notes":
Ashlander elders complain of prophecies which have been lost to tribal memory due to the carelessness or ineptitude of earlier generations of wise women and ashkhans. Suspicious scholars wonder whether these prophecies might have been deliberately forgotten or suppressed. Three Nerevarine prophecies in particular are said to have been lost: 1. The Lost Prophecies; 2. The Seven Curses; and 3. Seven Visions of Seven Trials of the Incarnate. Perhaps these lost prophecies will someday be found, either in forgotten accounts written by literate travelers, or in the memories of isolated Ashlanders, or in the secret traditions of the wise women and shamans.
The most direct description of the Nerevarine's goals are in The Seven Visions
Fortunately, all three lost prophecies are available in the game. The one most relevant to us right now is The Seven Visions, which had the following at the end:
His mercy frees the cursed false gods,
Binds the broken, redeems the mad.
He speaks the law for Veloth's people.
He speaks for their land, and names them great.
The game makes it clear that the "cursed false gods" are the Tribunal, Veloth's people are the Dunmer, and "their land" is Morrowind.
So did the Nerevarine fulfill the prophecy?
Yes. He ended the reign of the three cursed gods (Vivec in the main game, Sotha Sil and Amalexia in the Tribunal expansion) and spoke for Morrowind, being the representative of the three Great Houses and the four Ashlander Clans.
The common belief that the prophecy said the Nerevarine would "drive out the invaders of Morrowind, and shall reestablish the ancient laws and customs of the Dark Elven nations" does not match the actual prophecies. Sometimes people interpret prophecies to mean what they want it to mean.
As for the destruction of Morrowind, that was a clear consequence of overthrowing Vivec, as the floating Ministry of Truth was only being kept afloat by his power. The prophecy says nothing about what will happen after the Tribunal is defeated, so it is unrelated to the prophecy.
In other words, all the bad things happened because the Nerevarine succeeded in fulfilling the prophecies.