Could the Large Hadron Collider destroy the Earth?

No, it can't. The Large Hadron Collider (LHC) is a large (high energy) particle accelerator, and it is the most powerful one ever built on this rock. There is some talk about the LHC creating tiny black holes. The energy of the colliding particles will be really over the top, but a black hole is created when gravity overcomes all other forces to trap and compress a significant quantity of mass into a singularity. The energy of "creation" that appears when the LHC slams stuff together will be too high to allow the gravity associated with the events to actually create a black hole. And this takes into account the increase in mass associated with relativistic velocities.

This is similar to the "scare" associated with the test of the first hydrogen bomb (a nuclear fusion weapon). Someone ideated that the intense heat that would cause the fusion events to begin would result in the creation of so much energy that the atmosphere would begin the undergo fusion. This would cause the chain to go out of control and destroy the earth. This did not happen - because it could not. (The "geometry" of the blast is "wrong" for this to occur.) And the LHC creating those little black holes? Keep dreamin' about it. Some numbers on paper suggest things that reality won't support.