Would Gollum qualify, as a Ringbearer, to go to the Undying Lands?

Had Gollum not died at the Crack of Doom, it's reasonably likely that he would have died shortly afterwards. Note that Bilbo's age caught up with him after the One Ring's destruction;

...Bilbo was sitting in a chair before a small bright fire. He looked very old, but peaceful, and sleepy. He opened his eyes and looked up as they came in. 'Hullo, hullo!' he said. 'So you've come back? And tomorrow's my birthday, too. How clever of you! Do you know, I shall be one hundred and twenty-nine? And in one year more, if I am spared, I shall equal the Old Took. I should like to beat him; but we shall see.'

Since Smeagol was hundreds of years old, had he not committed suicide or dived into the fire in the hopes of rescuing the One Ring or been instantly destroyed by its destruction (as was Sauron), he would simply have just died of old age.

That being the case, a trip to the Undying Lands would have been out of the question.

I'm not aware of any canonical reasons why he would or wouldn't, so I'll draw on what we know about the Elves, Rings, and the Undying Lands. I seriously doubt they'd let Gollum join them for three main reasons.

The Hobbits were an exception
Until the end of ROTK, the Undying Lands were basically exclusively for Elves. But, as Gandalf hinted at, Bilbo was "meant" to have the Ring, which in turn meant that Frodo was "meant" to have it (and by extension, Sam, who bears it for quite a while in the book). But who intended for this to happen? Gandalf's hint seems to lead us to believe that some other force or power conspired to make it so. While not explicitly stated, it's within reason to say that Eru himself intervened and led Bilbo to find it. This is speculation, but I think it's within reason, seeing as how Gandalf has a knowledge of a great many things (and wasn't just trying to give a pep talk to Frodo). So, since the Hobbits were chosen by the Creator himself, they were granted admittance to the Undying Lands. Gollum wouldn't have been afforded that same benefit.

Gollum was already "stretched too thin"
After losing the Ring, Bilbo himself barely made it. Gollum was much, much older, and in much worse shape. Even if he made it to the Havens, and even if he made the journey, mortals still die in the Undying Lands (oddly enough). It just wouldn't be practical to fix up an express boat just to get him there before he kicked the bucket.

Gollum was inherently evil
I think this is the most important point. Notice how when Smeagol first laid eyes on the Ring, he instantly wanted it. Within moments, he succumbed to the Ring's influence and murdered Deagol. This isn't the work of an inherently "good" person. Frodo, Sam, and Bilbo all treated the Ring as a "precious" thing, but none of them killed over it. Frodo came close, but that was when the Ring was doing its utmost to dominate his will. For this reason above all others, I think the Elves would deny him asylum. You could even extend this logic to explain why the Nine, the Seven, or even Sauron himself would not be allowed to make the trip.

Again, this is just the best explanation I could come up with based on the established lore.

In Tolkien's retelling of the Judeo-Christian creation, the fall of man, and the reconciliation of the angels, the Havens represent Heaven and the Elves the fallen angels. The fallen angels are allowed to return to Heaven because they have reconciled themselves with God.

Given a concept of divine forgiveness, Gollum would be accepted into Heaven if he repented his evil ways.

LOTR is all about redemption, as has been said oft-times in answers to other questions on this SE. Gollum had the opportunity to act righteously, to assist the Fellowship to destroy the Ring (or at least not interfere), but instead, he chose to act in his own interest. Smeagol's struggle versus Gollum in the Marshes is a part of that drama. The reader gains hope that Smeagol may win out, and his soul be redeemed. The two are not separate beings, but one and the same, and so must ultimately be accountable for his choices.

I think Gollum/Smeagol is perhaps the greatest tragedy in LOTR. He succumbs and acts in self-interest, so he can never attain that which would have brought him peace. If he had made the better choice, to defeat absolute evil, I would say he would definitely be eligible for the Grey Havens -- maybe in a more agreeable physical form. He might die beforehand, but not because he's old. He was already ancient, many times the normal lifespan of a hobbit, so old age alone wouldn't be enough to kill him. He had already been without the Ring for decades, so another few years would be ok. What might do him in is that perhaps his soul was too tightly connected to the Ring. Once it was destroyed, he might not be able to maintain his corporeal self any longer.

I think that Gollum and Smeagol were two separate persons. The "character" the ring was capable of evoking in such creatures as Galadriel (instead of a dark lord you would have a dark queen) and Gandalf (through me it would wield too great a power) indicate that even they (Elflord and Maia wielder of Narya) would exhibit profound darkness if the ring were given the chance to do so.

If Gandalf and Galadriel are as susceptible to become dark rulers, then it is likely the person of Gollum is a manifestation of the ring itself through the being and person of Smeagol. I think that the degree of cunning and malice exhibited in Gollum shows the character of Sauron.

The construction shows that "into the ring was poured malice" and "will to dominate". Those are attributes of intentionality, and were done with purpose in aiding the ring of rings to achieve for Sauron what he intended.

The one ring is meant to be a ruling ring of rings. The rings were meant to cause leaders of a race to be able to evoke the most characteristic excellences of the races. The "center of elvendom on earth", Lothloriedorinian, came into being as it was in part by the use of the Elven rings. The great dwarven kingdoms likely have similar correlation. The ring was meant to make rings of rulership more rings of rulership - more capable.

The rings that were not hidden were poisoned by the one ring. The darkness poured into the one ring, leaked out through the others and into their bearers. Greed to the dwarves. I think that humans succumb more quickly to both the greatness and the darkness of the rings. Yeah - a fair bit of supposition. Gandalfs attribution of resistance to the ring by the Hobbits was due their race, not their character. They seem to be children of a different Valar - and as such not under the ring. I think Tolkien may have envisioned them as the "wee folk" of the Irish.

In the end, when Frodo fell to the ring, and claimed it for his own, it was the action of Smeagol that resulted in the destruction of the ring. It is a proof that evil, especially the character of Sauron could not live in a world with a copy of itself. It is a demonstration of the self-annihilation of evil. I think that, in a Tolkien world, while there is redemption, it is never cheap. Often, like the case of Grimma, it is accomplished at the cost of ones life. The destructiveness of even a little darkness is very great and it takes a lot of good blood spilled to quench it.

While I agree that the unnatural long life granted by the one ring evaporated as the ring did, I think in principle that Smeagol would have been welcome on distant shores. In an ironic way the thing that cost Smeagol his life, would have counted as sufficient for redemption.

EDIT: If being ringbearer alone was a qualification to go to undying lands would it not thus qualify both Isildur and Sauron to go there? I think there must be some additional or alternative criteria there.