Today, most college graduates have probably taken at least one class in world literature that involved reading some secondary sources of literary criticism.
My question is when literary criticism as a specific discipline began in the Western world.
The only relevant source I can think of off the top of my head would be Aristotle's Poetics; however, I consider this to more properly considered a work of aesthetics that focuses on why certain pieces of art (like the Iliad) are successful as art, rather than literary criticism that attempts to interpret a specific piece of art. Of course, if I am mistaken on this front, please correct me.
I suppose this depends really on what you mean by "literary criticism as a specific discipline". I would argue that literary criticism exists in Antiquity but under the species of rhetoric. Personally, I find James Turner's book, Philology helpful in tracing how the Humanities became disciplin-ified. In the case of literary criticism, he points to a shift in the early 19th century, from teaching rhetoric and oratory for hte purpose of improving compositional skills, to teaching litearture 'for its own sake'. For example, George Ticknor arrived in 1819 at Harvard and began teaching French and Spanish literary history and criticism.
The type of criticism you are thinking about is Evaluative criticism, and branches out of this philological background, paralleled by the emergence of Text Criticism and Literary History. By the mid 19th century you could begin to talk about Literary Criticism as a distinct discipline, but before that it exists subsumed under a broader field of 'Letters' or 'Humanities' or, as I said in Antiquity, you would just consider it part of the work of rhetorical analysis (So, for example Dionysius Thrax and Quintillian treat it as such).