There are some interesting side points to remember that relate to this question. Frodo's need for healing, both physical and psychological, is definitely the immediate factor that precipitated his departure to the West. However, if we read the book carefully, and look at Tolkien's earlier drafts in HoMe, something else also becomes apparent.
As far back as Tom Bombadil's house, Frodo is having dreams that foreshadow his later departure:
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That night they heard no voices. But either in his dreams or out of them, he could not tell which, Frodo heard a sweet singing running in his mind: a song that seemed to come like a pale lightbehind a grey rain-curtain, and grow stronger to turn the veil all to glass and silver, until at last it is pulled back, and a far green country opened before him under a swift sunrise.
The vision melted into waking....
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I love this passage!
These same words are used by Tolkien to describe Tol Eressea as the Elven vessel reaches its destination.
From the very beginning of the book, Frodo is drawn to Elvish things. He speaks at least some Elvish and has obviously gone with Bilbo on his nightime tramps in the woods near the Shire to find and see Elves. Also, throughout the book, there is a light in his eyes (similar to that of the Elves), which intensifies as the quest continues until it is finally quenched by the power of the Ring. It is Sam who is best able to see this strange light and comment on it: he clearly says this is the reason he loves Frodo, i.e. what that light represents.
And, in Rivendell, when Sam comments on how wonderful the place is and asks Frodo if anything is missing, the latter responds by murmurring 'The Sea...."
Even in the earliest drafts of LotR, when Frodo was still called 'Bingo', and long before Tolkien came up with the idea of Frodo being so damaged that he had to leave the Shire, Tolkien still had Frodo leaving. He says that Frodo withdrew to a small hut on the edge of Hobbiton and eventually left for the West, since he had become too "rarified" to stay in the Shire. (This was the word Tolkien used.)
So even from the beginning, before he is suffering from the guilt of having been unable to destroy the Ring and his continuing desire for it even after its destruction, Frodo was drawn towards the West and things Elvish. The amazing thing about Frodo is that for most of the Quest both his "bad" and "good" sides are growing at the same time. This is the only reason he's able to resist the Ring for as long as he did.
The hope is that the West will enable Frodo to reconnect with that "good" side, the part of Frodo that Gandalf alludes to while he sitting at his bedside in Rivendell:
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He is not half through yet, and to what he will come in the end not even Elrond can fortell. Not to evil, I think. He may become like a glass filled with claer light for eyes to see that can.
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This is a clear reference to the Phial of Galadriel, which contained a sliver of a Silmaril.
Given the fact that Frodo has faced ultimate evil, it is only by reconnecting with that type of light that healing may happen. And that is more possible in the context of the Blessed Lands than it would be in the Shire. As to whether or not that healing actually takes place, we do not know. Tolkien says that it is not certain if Frodo can be healed within the circles of the world.
Finally, one other possible impetus for him to leave.... Remember that Frodo tells Sam the person he most wants to be with in the world is Bilbo. (This point is often lost, especially with the movie images heavy on our mind.) I have to think that the presence of Bilbo made a difference in Frodo's decision to depart.
A thought way outside what the book actually tells us..... I have sometimes fancifully imagined that part of Frodo's healing in the West might be to serve and help his elderly uncle whom he loved so much, just as Sam had assisted Frodo on the trip and was continuing to do so in the Shire. Sam and Rosie were the ones offering their generous help to Frodo for the two years he did stay in the Shire. But perhaps Frodo needed to be in a situation where someone else he loved --in this case the elderly Bilbo -- truly needed his help. Sometimes going through the motions of helping someone will eventually catch hold in a person's heart and lead to healing and peace.