Here's how I answered that question back when I wrote my GMR review of the scores:
Why not use the Fellowship theme there? Or the Gondor Theme, since Aragorn muses on returning home and claiming his lineage? It turns out that Howard Shore knows his story well: from that point, only the Ring and the Ringbearer (and his unfailing companion, Sam) will go on; everyone else either dies (Boromir) or turns west into Rohan.
I still like that answer, at least in part: the LOTR scores are full of sophisticated instances of musical foreshadowing, such as the Gondor theme being heard during Boromir's speech at the Council of Elrond, one-and-a-half films before we ever even get to Gondor. But Scott provides his own answer:
While the motif does represent the Ring, it also represents all the lost lore/wisdom/culture of the previous ages. Man can no longer build things like the Argonath. The end of the Ring means the end of the Elves power and culture. I think it overall represents the transition from the Elf-dominated world to a new one, which will be dominated by either Men or Orcs. It also captures the tone of Tolkein's books, with a nostalgic focus on the glories of the past, never to be repeated.
I tend to be a bit more literalistic when trying to pick apart leitmotif-based scores, so I'm going to revise and extend my original remarks a bit. Consider what the Gates of Argonath actually are: while the film does not make this clear, they are not just two giant megalithic statues of generic kings. They are two very specific figures, actually: Isildur and Anarion, the two heirs of Elendil, the King who was slain on the fields at the Battle of Dagorlad when Isildur then took up his father's sword and cut the Ring from Sauron's hand. And Aragorn is passing through those Gates, not just as the leader of the Fellowship but as the Heir of Isildur.
But why the Ring Theme, again, as opposed to the Gondor theme, or the "White Tree Theme"? Because one of the things that Shore's scores do so well is musically illustrate relationships that exist between the various elements of the story (example: the motifs associated with Saruman and with Gandalf the White both start with the exact same notes). In this case, remember the scene early in Fellowship when Gandalf travels to Minas Tirith to consult the account of Isildur, from which he reads the following:
"It has come to me -- the Ring of Power! It shall be an heirloom of my Kingdom...all those who follow in my bloodline shall be bound to its fate, for I will risk no hurt to the Ring...it is precious to me, though I buy it with great pain..." (emphasis added)
So when the Fellowship passes the Gates of Argonath, the Ring is passing the likeness of the King who took it for his own, and who used it to build his Kingdom, and who was betrayed by it to his ruin; and the Ring is accompanied by that King's heir. I'd say that makes the Ring motif directly relevant to the Gates of Argonath.
Of course, Scott is right that it's sometimes hazardous to assume a one-to-one relationship between leitmotifs and the things they represent; the proper use of leitmotif is, as I am always pointing out to people on various film music forums, more than just a simple matter of a composer saying, "Oh, Bob is on the screen now, so I must play Bob's Theme". But I constantly find solid, storytelling reasons for the gestures that Howard Shore makes in the course of these three filmscores.
(By the way, some of my very first thoughts on the LOTR music, based only on Fellowship because that was the only one in release at the time, can be read here, in a post that actually originated on Usenet but which I then re-posted to the blog for my own reference purposes.)