In The Lord of the Rings: The Return of the King, just after the One Ring was destroyed in Mount Doom along with Gollum, there is a scene in which Frodo says:
Sam, I could not have destroyed the Ring.
Why did he say that? Why was he unable to destroy the Ring?
The complete quote from the book is:
'Yes,' said Frodo. 'But do you remember Gandalf's words: Even Gollum may have something yet to do? But for him, Sam, I could not have destroyed the Ring. The Quest would have been in vain, even at the bitter end. So let us forgive him! For the Quest is achieved, and now all is over. I am glad you are here with me. Here at the end of all things, Sam.'
So, that line is supposed to mean that Frodo recognizes that in a way it was only by fate that The Ring was destroyed.
Remember:
When the time came to actually drop the ring in the fire of the mountain, Frodo was on the point to fail, refusing to do what was needed. In the end, it was only because of Gollum that the ring was actually destroyed. If Gollum hadn't bitten Frodo's finger off and fallen into the volcano with the ring, the mission could have ended in a failure.
With that phrase Frodo is acknowledging that.
..and also saying that in the end Gollum fulfilled his last part in the story, just like Gandalf said he would.
As for the second part of the question - why Frodo wouldn't want to destroy the Ring after going on such an hard journey to reach Mount Doom... that is actually one of the central plot devices in the books. Without going in great detail, the Ring has the power (and the will) to corrupt its current bearer, and its hold on the bearer will only grows in power with time and its vicinity to its true master, Sauron. That is why when confronted with the final choice to destroy the Ring once and for all even Frodo in the end fails and only fate saves the day in the form of a creature that was once spared out of pity and now finally plays his last part in the story.
For more details about the Ring influence, have also a look at this question:
The Ring protected itself. Each person who saw the Ring was drawn to it, sometimes quite out of character.
Isildur took the Ring from Sauron's hand and, even though he knew what it was and what harm it had done and with Elrond counselling him to destroy it, could not bear to destroy it which, physically, would have been easy, since they were already on the slopes of Mt Doom. He writes:
But for my part I will risk no hurt to this thing: of all the works of Sauron the only fair. It is precious to me, though I buy it with great pain.
Smeagol sees it and immediately lusts after it, kills his friend to get it and then hides in the mountains for many centuries muttering about his "Precious".
Bilbo was nearly unable to give it to Frodo (even with Gandalf's help). Destroying it would have been much harder.
The Ring takes over:
And if he often uses the Ring to make himself invisible, he fades: he becomes in the end invisible permanently, and walks in the twilight under the eye of the dark power that rules the Rings. Yes, sooner or later - later, if he is strong or well-meaning to begin with, but neither strength nor good purpose will last - sooner or later the dark power will devour him.'
and
He said that [the Ring] was "growing on his mind", and he was always worrying about it; but he did not suspect that the ring itself was to blame.
Gandalf says:
A Ring of Power looks after itself, Frodo. It may slip off treacherously, but its keeper never abandons it. At most he plays with the idea of handing it on to someone else's care - and that only at an early stage, when it first begins to grip.
And if it won't let itself be abandoned, it even more strongly resists being destroyed. Once he learns of the Ring's evil, Frodo asks Gandalf:
'But why not destroy it, as you say should have been done long ago?' cried Frodo again. If you had warned me, or even sent me a message, I would have done away with it.'
'Would you? How would you do that? Have you ever tried?'
'No. But I suppose one could hammer it or melt it.'
'Try!' said Gandalf. Try now!'
Frodo drew the Ring out of his pocket again and looked at it. It now appeared plain and smooth, without mark or device that he could see. The gold looked very fair and pure, and Frodo thought how rich and beautiful was its colour, how perfect was its roundness. It was an admirable thing and altogether precious. When he took it out he had intended to fling it from him into the very hottest part of the fire. But he found now that he could not do so, not without a great struggle. He weighed the Ring in his hand, hesitating, and forcing himself to remember all that Gandalf had told him; and then with an effort of will he made a movement, as if to cast it away - but he found that he had put it back in his pocket.
Gandalf laughed grimly. 'You see? Already you too, Frodo, cannot easily let it go, nor will to damage it. And I could not "make" you - except by force, which would break your mind.
When Sam and Frodo reach Mt Doom, Frodo is unable to destroy the Ring:
'I have come,' he said. 'But I do not choose now to do what I came to do. I will not do this deed. The Ring is mine!'
And after Gollum has destroyed it, he says:
'But do you remember Gandalf's words: Even Gollum may have something yet to do? But for him, Sam, I could not have destroyed the Ring.
