Which arguments can prove that the Greeks and Romans did not practice linguistics in its modern meaning?
I would not know what you call linguistics in its modern meaning (probably not the same thing I would). Linguistics has so many aspects. Were grammarians in former centuries doing linguistics? What of Alexandrine grammarians, though their work was not quite similar to later grammarians a millenium later (and some). What of these later grammarians ? Grammatical concepts evolved slowly and I suspect (I am only an amateur) often with prescriptive rather than descriptive intent at first (e.g. the Académie Française in the 17th century).
Neverthelees, the Greek civilisation was very adept at analysing texts and discourse (is that modern linguistics?). It actually lead to the discovery of logic and rhetorics. And logic happens to be now, in various guises, a fondamental ingredient of language analysis.
They started things, and building up concepts and organized thought and knowledge is a very long process of evolution. This brings questions on the structure of this evolution, But it is probably not the place to ask.
Were the Greeks doing modern physics. Well, they knew some laws. Does that qualifies them even though they knew nothing of the modern description of the world, of the standard model, of atoms and electromagnetism, or of gauge theories. Archimedes is indeed considered a great physicist.
I just found by chance a very relevant reference, though Indian rather than Greek or Roman. According to Wikipedia, the Indian Sanskrit grammarian Pāṇini did very early work in the 4th century BC, that apparently had a strong influence on modern linguistics. Still according to Wikioedia, Pāṇini's work is purely grammatical and lexicographic (Sanskrit morphology).
In his monograph Mathematical Linguistics, András Kornai starts the foreword as follows:
Mathematical linguistics is rooted both in Euclid's (325BCE 265BCE) axiomatic method and in Pāṇini's (520BCE 460BCE) method of grammatical description.
Pāṇini's contribution was first noted here by Gaston Ümlaut, in a comment above.
So that may be another proof that our early forefathers did real science, ... and that it was not limited to Greece or Rome and the Mediterranean area as western culture would sometimes have it.