Yes, generally speaking, the "scientific method" could be approached epistemologically. Epistemology, simply put, concerns how we know things and how we can validate judgements. It is one of the main branches of "metaphysics," along with ontology (what things exist), logic (correct analysis of propositions), and ethics (correct practices).
Certainly, epistemology would play a central role in a philosophical inquiry into the scientific method. A good example is how we deal with instrumentation. Strange as it might seem now, there were arguments in Galileo's day about what exactly one was seeing through a telescope. Obviously it is a distortion. So, is the very different appearance of the moon "more real" or "less real"? How can we interpret those shadowed crinkles as mountains?
However, epistemology as a field is rather broad, not limited to science, and practicing scientists are not overly concerned with it. I would not start out looking under "epistemology" in the card catalogues or databases. You are better off starting out directly with introductions to the philosophy and history of science. I stress "history" because this is actually one of the better ways to understand how the scientific method has been understood and how it has evolved in history.
Some key historical moment would be Francis Bacon's original case for empirical verification and the "inductive" method, later institutionalized in the Royal Society. Here we have the basis of direct observation and publicly demonstrable "experiment," as developed by Galileo and many others. There ensued a veritable mania for cataloging, experimenting, demonstrating, and recording often with very little in the way of overall theory about what science was actually doing.
Next, I would look at the fascinating literature surrounding Newton's theory of gravity, momentous yet more controversial than we now recall. Here the role of mathematics takes central stage. And Newton's enormously decisive step in demonstrating how this invisible thing or "force" he names "gravity" works, while refusing to say what it is. To many scientists of his day this "gravity" looked like today's quantum entanglement or "spooky action distance," a dangerous slide back into metaphysics. Yet it worked. In this way, by simply ignoring ontology, Newton liberated physics from metaphysics by division. Or at least partially.
From this point, we get into issues of induction, causation, evidence, etc., and Popper's highly influential description of scientific method as induction and falsification. Science can never absolutely prove what is true, but only falsify what is not. A process of hypothetical, public conjecture and refutation. While oversimplified, this is a crucial distinction often lost upon the public.
After Popper come those "post-modern" turns that are both sociological and more overtly epistemological, viewing science as a kind of social consensus that can radically evolve and perhaps has no fixed standard. Kuhn and Feyerabend are the best known names is this then "radical" reevaluation of the many "methods" used by scientists. Of course, many other names, such as Hempel, Carnap, or Sellars, have much to say about scientific method.
In terms of "educated in epistemology" certainly it doesn't hurt. But the issues implied in your question may have more to do with a good general understanding of scientific method...and then a grasp of statistics and probability, which are so often misunderstood in evaluating an publicizing findings, even by scientists. Epidemiology is a good topic here.
The literature is vast, so just googling should lead to many relevant works. My only possibly "unique" suggestion is not to neglect the history, for with each great theoretical shift, from Newton to Darwin to Maxwell to Einstein, "scientific method" also changed. One addendum. Remember, because others will say otherwise, that studying "the scientific method" is both history and metaphysics. Many philosophers of science dismiss "metaphysics." But to say, for example: "Physics is our best method of understanding reality" is not a statement provable by physics. It is a metaphysical statement.
