'This task is appointed to you'

[Morgul Queen]

Isn't the actual quote (book-wise) 'This task has been appointed for you...' which basically, when told to Frodo, means that although he isn't forced to do this thing, he is meant to do it, it is there *for* him to do...if he wishes.

This is further elaborated on when Sam later talks about the 'Great Stories' and how in the best ones - the most meaningful - the Hero gets plenty of chances to turn back, to turn away from the quest that he has set out from, e.g. Beren going to get the Silmaril, Luthien going after Beren, Tuor trying to find Gondolin, Earendil searching for his parents and Valinor, Elwing searching for Earendil, most of thses are done out of love for another, it is the same with Frodo. He continues on, not for a romantic love but a deep abiding love of the Shire and its people, Sam continues on for Frodo. And in all these quests it is love that sees them through.

When all is said and done, Frodo is so deeply changed by what he had to endure to see the quest through that he can no longer rest in the Shire, he has almost outgrown it, you could say. In the same way a 10 year old cannot truly find peace in a Kindergarten, Frodo needed to 'Graduate' to a place where his inner-most needs could be recognised.