The One Ring extended Gollum/Sméagol's life and vitality by 478 years and Bilbo's vitality by 60 years. Did it have the power to make them truly immortal if they had kept it, or did it just grossly slow down the aging process? The ring was made by a Maia for a Maia (spirit), not a mortal.
A mortal wearing a ring of power does not age, nor can he die of natural means (Gandalf says this in FOTR when he tells Frodo of the rings of power)
For evidence see the Nazgul, who were men that lived in the middle centuries of the 2nd age, they survived until the end of the 3rd age after being gifted rings of power, a period of around 4-5 thousand years.
Also see Gollum who lived 500 years after taking the ring.
However, whether the rings were capable of immortality without corruption is debatable, I find it hard to believe that the rings were ever created with the intention of corrupting men to evil (Sauron was involved in their creation (except the 3 elven rings)) but I think that it was the creation of the One that corrupted the mortals wearing the rings.
That said the dwarves were originally made by Aule to be resistant to corruption and the devices of Melkor, and as such they resisted the rings more than men, and did not recieve immortality from them. However the Dwarven rings corrupted them with their specific weakness (Gold lust).
Finally any mortal wearing a ring could still be killed, and regardless of how long they lived and their power they would still go to their respective fate. Men (hobbits as well) would go to "Mans Doom" and leave Arda, and Dwarves would be gathered by Mandos.
The fact that a man with a ring of power can be killed is evidenced in several places:
So in answer to your question:
Immortality in Tolkien is very different to not dying.
To be immortal means that your spirit is bound to the world, the Music of the Ainur is as fate to it, and it does not leave the world until the end of the world, beyond which its not known what happens to it.
To be mortal means that your spirit is not bound to the world, it can shape it's own destiny beyond the Music, but it must die and leave the world after a short time; beyond the end of the world it is destined to take part in the Second Music of the Ainur.
Tolkien is quite clear in several places that only Eru can change a being from mortal to immortal or vice-versa, for example:
Therefore the One Ring is quite incapable of making a mortal creature become immortal.
This can be seen in Gandalf's words to Frodo concerning the effects of the Rings on mortals in Shadow of the Past (my emphasis):
And is completely consistent with statements elsewhere, for example the following footnote to Letter 131:
A mortal who uses the Ring is therefore still a mortal: merely one who's lifespan has been prolonged (indefinitely and intolerably), but they are still subject to the fate of mortals, which remains completely unchanged.