The real reason for Gollum's death

There is a short, lesser-known passage from near the end of the Return of the King, the last of the Lord of the Rings books, which suggests that Gollums death may not have been accidental. Instead, Gollum's fall into the Crack of Doom may have been the direct result of a magical death curse laid on Gollum by Frodo, shortly before their last battle.

Here is the where the quote begins:

On the slopes of Mount Doom, Gollum had tracked Frodo and Sam there and made a last-minute grab for the ring. "Down, down!" frodo gasped, clutching his hand to his breast so that beneath the cover of his leather shirt he clasped the ring. "Down you creeping thing, and out of my path your time is at an end you cannot betray me or slay me now!""

Then suddenly, as before under the eaves of the Emyn Muil, Sam saw these two rivals with other vision. A crouching shape, scarcely more than the shadow of a living thing, a creature now wholly ruined and defeated, yet filled with a hideous lust and rage. And before it, stood stern, untouchable now by pity, a figure robed in white. But at its breast, it held a wheel of fire. Out of the fire there spoke a commanding voice: "Be gone and trouble me no more! If you touch me ever again you shall be cast yourself into the fire of doom!".

Frodo then seems to come out of a trance, gasping. Gollum shrinks back and Sam dashes in to protect his Mr Frodo.