Men were never allowed to enter the Undying Lands after the fall of Numenor and any attempts to reach the lands would result in failure or disaster. Aragorn, despite his heroic deeds, likely wouldn't have accepted an invitation to ensure his duties as king.
Why can't Arwen go to the Undying Lands after Aragorn dies?
Only Elves and those deemed worthy (like Gandalf and Frodo) could sail to the Undying Lands. And by choosing to stay with Aragorn, she was forgoing her place on the boats and binding herself to the realm of men, thus becoming mortal.
After Aragorn's death, Legolas made a ship in Ithilien and left Middle-earth to go over the sea. His strong friendship with Gimli prompted Legolas to invite him to accompany him to the Undying Lands; making him the first and only Dwarf to do so. He was never seen again in Middle-earth.
The Undying Lands are a place of extreme blessings, essentially the heaven counterpart of Middle Earth. Very few are permitted to go there after an ancient rift in which some elves chose immortality, and others chose to take on a mortal lifespan.
Gandalf's last job in Middle-earth was to bring Bilbo and Frodo Baggins, the Ring-bearers, to the Undying Lands. It could be that this task was exactly why Gandalf stayed in Middle-earth for so long after the One Ring was destroyed.
Why Couldn't Men Live in the Undying Lands? | Lord of the Rings Lore | Middle-Earth
Would Frodo be immortal in the Undying Lands?
The purity of Valinor gives Frodo relief from his wounds since the presence of evil in Middle-earth keeps them fresh, and he likely lives a considerably longer life than he would have. Still, he doesn't live forever.
Yes. In appendix B of the Return of the King, there is a chronology of some events that took place after the story, and that includes a few times Sam, Merry, and Pippin met Aragorn.
After the death of King Elessar, Legolas made a ship in Ithilien, and through Anduin, he left Middle-earth to go over the sea. His strong friendship with Gimli prompted him to invite Gimli to go to the Undying Lands; making him the first and only Dwarf to do so.
Merry and Pippin were well-known for hosting great parties with songs and stories all throughout the Shire. Then, of course, when Frodo chose to go to the Undying Lands with the Elves, Merry, Pippin, and Sam accompanied him to the Grey Havens to say their goodbyes to him and Bilbo.
Elrond confronts Aragorn in The Return of the King and says that Arwen is dying, her fate tied to the Ring. Arwen isn't necessarily dying because of the Ring, but now that she is mortal, she is dying through the slow decaying of time.
Arwen reciprocated Aragorn's love, and on the mound of Cerin Amroth they committed themselves to marrying each other. In making that choice, Arwen gave up the Elvish immortality available to her as a daughter of Elrond, and agreed to remain in Middle-earth instead of travelling to the Undying Lands.
Did Galadriel refuse the Undying Lands in the books?
After Morgoth's defeat at the end of the First Age, elves like Galadriel were pardoned and offered a return to the land they abandoned. Galadriel refused, partly because she believed her people had no reason to need pardoning, but more because she always wanted to witness Middle-earth and rule a kingdom there.
After his wife's passing, Sam entrusted the Red Book of Westmarch to Elanor. According to Elanor, Same went to the Grey Havens and sailed to the Undying lands to be reunited with Frodo. As Sam was also a Ring-bearer (although only for a brief time) he was granted passage to sail into the West.
According to one theory (via Stack Exchange), Gimli did not know about the fall of Moria in The Lord of the Rings: The Fellowship of the Ring movie due, effectively, to poor communication.
Gimli is not the last of the dwarves. This is a common misconception because Gimli is the only dwarf who gets a fair amount of coverage in the franchise. Lord of the Rings focuses its narrative primarily on the affairs of men, elves, and hobbits. Dwarves are a secluded race that lives in the mountains.
Gimli didn't know about Moria's fate because there had been no communication between Erebor and Moria for 25 years, and the Dwarves were hopeful and did not want to accept failure until there was certain news.
At the age of 102, Merry returned to Rohan and Gondor with Pippin; they died in Gondor, and were laid to rest among the Kings of Gondor in Rath Dínen, the street of tombs, then moved to lie next to Aragorn.
Legolas stuck around for another 120 years before Aragorn died, which was when he finally decided that his own time in Middle-earth might be up as well. He built his own ship and, in a pretty unprecedented honor, invited Gimli to accompany him to the Undying Lands.
Legolas does not marry or have any children, based on the appendix of The Lord of the Rings. Regardless of who his mother was, Legolas is the last of his line. When he sails into the West at the beginning of the Fourth Age, he leaves no family members behind.
Unfortunately, as you've probably guessed, that's not the case. Frodo, Sam, Bilbo, and Gimli all go West to die. Living in Aman extends their life somewhat, so Sam could have met Frodo one last time, but this is not the happy ending you think it is.
Once an Elf is granted passage from the Grey Havens to Valinor, they sail across the Sundering Seas and enter the Undying Lands, never to return to Middle-earth. Valinor is its own sort of heaven, an eternal land of paradise for Elves to live among the gods.
Not only are dwarves prohibited from entering the Undying Lands, but Gimli was never a Ring-bearer, therefore has no obvious right to be accepted into the Valar's home. Perhaps the blessings of Legolas, Gandalf, Elrond and Galadriel was enough to get the dwarf a ticket into the party.
