I thought that Gimli was also so that he could live the rest of his life with his new friend If not where did he go ?
If that was the case then why couldn't the other members of the fellowship have the privilege to do the same as Gimli? Obviously we know Frodo, Bilbo and Samwise had that gift, but for different reasons completely. In the 1st Age there were many greater Elves than Legolas and they befriended men who were lords of men of the Edain, whom did as great if not greater deeds than Gimli, and they were still not allowed access to the undying lands. I just don't see why Gimli should be allowed the grace of the Elves and Aragorn, Túrin, Húrin, Beren, etc... (The list go on for quite some time) were not granted access.
I believe after Gimli died he would have gone to the halls of his fathers and dwell there until after the Final Battle was over and the dwarves helped to rebuild Arda.
It is also stated at the end of the Akallabeth,
And tales and rumours arose along the shores of the sea concerning mariners and men forlorn upon the water who, by fate or grace of favour of the Valar, had entered in upon the Straight Way and seen the face of the world sink below them, and so had come to the lamplit quays of Avallone, or verily to the last beaches on the margin of Aman, and there had looked upon the White Mountain, dreadful and beautiful, before they died.
We all know Dwarves are not fond of forests, and i doubt water, so in that case I think it's safe to say almost all the Dwarves never tried to sail the seas. Although Tolkien wisely uses the word 'mariners' along with men, even though Gimli was one of these mariners and sailed to Valinor with Legolas... Well the end of the quote can provide your answer as to what happens to him.