What is the long-term idea of the Ruy-Lopez opening? I have studied the London opening system on lichess.org. What really clicked with me are the systems two main ideas: (1) develop a defensive structure which (2) allows you to crash black's kingside castle by pushing pawns with rooks behind them (and very often crush your opponent).

What you have is plan A, which is to crash black's king side, and if black doesn't castle you return to plan B, which is to improvise from a very strong defensive posture. The main point is that you have plan A with a very specific way to win in mind.

I am /not/ asking about opening principles. I am asking about the specific ideas of the Ruy Lopez. I understand how it gets your pieces out fast, is aggressive, and allows for a fast castle. But what approach or strategy does the Ruy Lopez aim for? 

In my own games, the best idea I have is to position my rook in the center file (because I can castle quickly) and try to open up the center and use a discovered attack on the king or queen by exposing my rook. But is this 'classical'?
The answer to this is way beyond my pay grade, but I may be able to say something that prepares you for a better answer.

You are not only comparing two very different opening systems (London System and Ruy Lopez), but you are comparing two different ideas of what an opening system is. The London system was once described by GM Julian Hodgson as the "businessman's opening", ideal for those with little time for study, because once you have played d4 and Bf4 there is not much choice for the other pieces. It can be said without gross exaggeration that the London system can be learned in five minutes.

The Ruy Lopez is much more vague and much more ambitious. The vagueness is why no short answer is possible. The goal is to increase Whites mobility and control of space and to retain the option of where to strike. There are some standard manoeuvers such as c3 and d4,0-0 and Re1,Nb1-d2-f1-e/g3, a4, and others, but Black can often play a big role in deciding the sort of game it will be. He can go for complicated counterattack with the Schliemann, the Open, or the Marshall. He can play more safely, with Steinitz/Steinitz Deferred or more classical approaches like the Chigorin, Keres, Breyer.. In fact one reason NOT to play the Lopez is all of the work required to prepare for everything that might happen. Mastering the Lopez is a lifetimes work.