How does Buddhism refute the inherent existence of sub-atomic particles?
In this paragraph:


  Postulating the existence of a partless particle that would be truly
  singular will not work. It would be impossible to ... Having no
  differentiable sides, ... Such a particle would not be able to ... All
  the surrounding particles would ... Nothing with any extension could
  ever come to be.


The author is, in my estimation (I have no familiarity with the book or the author), doing one of two things:


Confusing Buddhism with Physics. Whether this particle is this or that particle is that, it won't help you become a better person and understand your mental reality any better. Although the above quoted paragraph would make an interesting question on physics.stackexchange.com.
Speaking poetically in order to help the reader become absorbed with the idea. Perhaps this is the case.


During an course I attended under the name "Introduction to Asian Philosophy", my professor explained emptiness as a flashy way of reminding us that we only define things in relation to what they are not. Nirvana is defined as everything Samsara is not, therefore Nirvana is "empty of essential qualities". Similarly, Samsara is meaningless without a Nirvana to compare it to. This logic can be extended to all entities and concepts: Without night there can be no day, without a whole there can't be parts, etc.

You may or may or not you agree with this reasoning, but for me it's a practical interpretation of the doctrine of emptiness that helps us to not cling to our perceptions of what things are.

In other words, as R. Barzell says in the comment below, Is emptiness ontology, or a way of seeing to free us from the bondage of concepts? Is treating emptiness as ontology missing the point?

Lastly,


  What are the other "practices capable of revealing the emptiness of matter even at the most basic levels?" Specifically this belief in the inherent existence of the "element of earth", i.e. sub-atomic particles.


Meditation will probably help, as it gradually helps the practitioner become free from the limitations of language, thoughts, and concepts. The better we understand the way our minds work, the less we can be obstructed by our minds.