While the other answers already point out both the technical basis for Roark's claims as well as a general cultural precedent for disregarding the blind following of tradition for tradition's sake in favour of doing what's "right", there is an even more direct historical basis for Roark's broader sentiments against blindly replicating the replicas of Greek monuments if we look at the emerging American architecture around the turn of the century -- especially two individuals therefrom -- which the book draws a lot of inspiration from. In fact the broader historical precedent permeates the primary artistic conflict throughout the story and its specific inception is hinted at not too much later in Chapter 3 when Henry Cameron is introduced.
But first let's make an excursion into that time at the end of the 19th century in our own world, specifically to the place that can be regarded as the breeding ground for modern American architecture: Chicago. The great fire of 1871 "freed up" a lot of space for an upcoming generation of architects to realize a new Chicago, together with the new possibilities for height that the concept of steel framing brought. Among the most prominent of those early figures would certainly be Louis Sullivan, often seen as an inspiration for Roark's mentor Henry Cameron. And while Sullivan's position as a truely "modern" architect could be debated in today's understanding1, he was nevertheless instrumental and influential in shaping not only Chicago but the modern American city (and isn't without reason called the "father of skyscrapers", not dissimilar to Cameron), yet supposedly swallowed by a historicising movement that also swallowed Cameron.2
When Cameron is introduced, his downfall is brought in direct connection to the World's Columbian Exposition in 1893, which was basically the World's Fair but with the added claim of celebrating the 400th anniversary of discovering the "New World", held nowhere else than in Chicago. The most prominent feature of this exhibition was the shaping of the "White City" (not to be confused with the ones in Tel Aviv and Gondor), an extensive complex of buildings called that for its literally whitewashed neoclassical architecture, a style that deliberately seeks to replicate the grandeur of the classical antiquity with its "famous columns and triglyphs".
While the White City and Columbian Exposition was instrumental in advancing not only the city of Chicago and its architecture but also ideas of urban planning, it wasn't without criticism, specifically the neoclassical style it propagated. And while Sullivan himself, like Henry Cameron, was a critic of this neoclassical approach3, it was primarily a young generation of architects that didn't see much sense in replicating these old European buildings instead of creating something new, befitting which is also the fact that many of the buildings were indeed made from whitewashed plaster copying "copies in marble of copies in wood", at the most extreme maybe more akin to a temporary film set than genuine buildings. Add to this that this idea of copying the "Old World" (itself copying the old Old World) could in fact be seen as a slap in the face of the Columbian Exposition's intention to represent the New World. The novel echoes some of these sentiments, especially with regards to Cameron's fate:
The Rome of two thousand years ago rose on the shores of Lake Michigan, a Rome improved by pieces of France, Spain, Athens and every style that followed it. It was a "Dream City"4 of columns, triumphal arches, blue lagoons, crystal fountains and popcorn. Its architects competed on who could steal best, from the oldest source and from the most sources at once. It spread before the eyes of a new country every structural crime ever committed in all the old ones. It was white as a plague, and it spread as such.
A country flung two thousand years back in an orgy of Classicism could find no place for him and no use.
And making the turn back to this idea of a New World, the most prominent of this young generation of architects would certainly be Sullivan's apprentice Frank Lloyd Wright, who was the inspiration for Howard Roark himself in many aspects (albeit not all). With the advent of the new century Wright departed from his mentor's firm, ultimately forming the Prairie Style, largely regarded as the first truely North American architectural style, supposedly devoid of European baggage5 (which was also pretty much the intention behind it as being fit to the American Midwest prairie). And while strongly related to the Arts and Crafts movement and one might at first hesitate to directly call this "modern architecture" yet (albeit being a rather fuzzy term in itself) when comparing it to later movements like Bauhaus and the International Style, it was already exhibiting a lot of its core principles6 and is thus very much early modernist architecture. In his 1901 lecture "The Art and Craft of the Machine" (as also part of Modern Architecture, Being the Kahn Lectures for 1930) Wright echoes not only Roark's sentiments (with a not too dissimilar rhetoric) but also the state of education at Stanton Institute of Technology as depicted in that same conversation with the dean (hereby the Chicago Art Institute, though). To quote just a few passages:
...Yet, why should an American university in a land of democratic ideals in a machine age be characterized by second-hand adaptation of Gothic forms, themselves adapted previously to our own adoption by a feudalistic age with tools to use and conditions to face totally different from anything we can call our own? [...] Nine out of ten public buildings in almost any American city are the same.
...These students at best are to concoct from a study of the aspect of these blind reverences an extract of antiquity suited to modern needs, meanwhile knowing nothing of modern needs, permitted to care nothing of them, and knowing just as little of the needs of the ancients which made the objects they now study [...] And this is an education in art in these United States.
...You are sunk in “Imitation.” [...] a general cheap machine-made “profusion” of -- copies of copies of original imitations. To you, proud proprietors -- do these things thus degraded mean anything aside from vogue and price? Aside from your sense of quantitative ownership, do you perceive in them some fitness in form, line and color to the purposes which they serve? [...] If not, you are a victim of habit, a habit evidence enough of the stagnation of an outgrown Art...
And this also brings us to another important idea that is inherent to Howard Roark's quoted criticism, namely that the materials and architecture of a building should fit to its purpose, echoing Sullivan's idea that "form follows function", as well as its environment, echoing Wright’s idea of Organic Architecture. Because Roark's argument isn't just that copying past cultures' works inhibits our individual creativity, but more that this copy misses the original purpose of the architecture it copies as well as that of the building it's trying to create, the same as the Greeks supposedly already missed the purpose of the wooden columns and triglyphs when they were copying them into marble (not to speak of the later generations copying those copies). Or as DQdIM also puts it in his comment, it compromises the building's integrity.
So to bring all these ramblings back to The Fountainhead and the inspiration it draws from the Chicago architecture scene, specifically the likes of Louis Sullivan and Frank Lloyd Wright7, the argumentation employed by Roark in this scene is directly referencing this revolt against empty replication of the old with neoclassicism without respecting the purpose of architecture, that was emerging at the turn of the century. While Chicago and the US weren't necessarily alone in these efforts, specifically in Chicago it ushered in a truely American and early modernist architecture propagated by noone else than Howard Roark's real-life inspiration.
Though, we have to note that Ayn Rand is about one generation late with this revelation when having it happen in the 1920s in the novel, when neoclassicism had already fallen quite out of vogue, the Prairie School was declining already too8 and post-WWI Europe was picking up its legacy (or incorporating it into its own modernist aspirations). Roark's style is also more reminiscent of Wright's (and others') later work from around the time when she wrote the book, yet clearly building on and referencing an architectural philosophy emerging in those earliest days. Likewise does she exaggerate the neoclassicism and the buildings it produces to almost caricatural heights, as well as the dogmatism with which it supposedly crushes any other way of thinking. But even if strongly inspired by the personalities and work of specific individuals, it is ultimately neither a history book nor a biography rather than a novel seeking to transport its ideas through a more generalized look at architecture. And even in real architecture the conflict between modernist up to postmodernist approaches and more conservative and historicising ones isn't a one-way street either rather than repeating itself in various forms.
1) While himself part of a broader generation of American architects trained at the École des Beaux-Arts in Paris, he was nevertheless eschewing excessive neoclassicism, while still not entirely avoiding "ambitious ornamentation". He has an interesting place in the transition from classic to modern.
2) Though, with all romanticising the two architects' supposedly similar fates we still have to keep in mind that Sullivan's decline can also be attributed to economical reasons and might not be solely to blame on a cultural mood shift
3) But in contrast to Cameron, Sullivan did not refuse to contribute to the Exposition and instead actually contributed the only not purely white neoclassical building.
4) This sarcastic use of "Dream City" can't help but evoke the image of Thomas Cole's The Architect's Dream, even if likely a nightmare for Henry Cameron.
5) It is of course debatable how truely original anything can be, especially in a culture born out of European colonies, and considering in turn the influences of Japanese art that Wright brought into his work. Nor is this necessarily a bad thing.
6) E.g. opening the floorplan, elongated window bands for natural light, concrete as a cheap and flexible material, embracing the building materials rather than hiding them, clear geometric shapes without superfluous ornament.
7) Of course the architectural landscape of ~1900 USA, including the ones daring onto new grounds, consisted of more than those two figures, but not only are they the most recognized from that era, they are also widely understood and documented as the direct inspiration for Henry Cameron and Howard Roark respectively.
8) And when Wright himself was delving into an even more unconventional architectural approach, specifically the Los Angeles textile block episode, ironically not entirely devoid of ornamentation and the not always unintended or non-stereotypical influence of earlier styles, albeit this time pre-Columbian Meso-American.
