First off, "harmony" to the ancient Greeks had a broader (and somewhat more specific) usage than in the question. Being "in harmony" to the Pythagoreans meant explicitly in a relationship of "harmonious number," but it didn't necessarily refer to notes sounding at the same time, certainly not polyphony.
The Greeks found their "harmony" among notes in relationships among the notes of the scale and in melodic relationships. This definition of "harmony," including melodic intervals, appears for thousands of years in music history, including throughout the renaissance.
So, it didn't take a millennium to use a 3:2 or 4:3 interval in "harmony." The Greeks, in their understanding of harmony, were already doing it.
But I know the question is asking about polyphony too. As other comments and answers have noted, it's likely that much homophonic/polyphonic music wasn't written down, even if it happened. It's also possible (perhaps likely) that besides the potential of polyphony, ancient Greeks sometimes made use of drones in their performances, which would highlight intervallic relationships between the drone and notes of the scale. That would likely involve 3:2 and 4:3 intervals among others.
However, I sense in the question a sort of notion that polyphony should be "obvious" or at least a "natural development" in music. There's a long history of Western musicians infusing the leap to polyphony with a notion of "progress" in music history, often also unfortunately viewing world musics that are monophonic or heterophonic as less interesting or even less "evolved."
The fact is that full-blown Western-style polyphony is not an obvious or common development in music around the world. It wasn't in Western culture, either, where it may have been an aberration until the 1500s or so when the printing press made the possibility of spreading polyphonic written music easier. (Prior to that, Western polyphony only tended to occur among elite circles, nobles who could pay for a small choir of trained singers, etc. It's possible, of course, that polyphonic or at least homophonic works were also being performed by common people, but we have very little written evidence one way or the other.) Polyphony only seems pervasive now due to the spread of global pop -- usually based on Western models -- in the past century.
Most traditional music is primarily monophonic or heterophonic in performance (with or without accompanying drones). That does not mean that such music isn't interested in the construction of scales -- to the contrary, many cultures that don't emphasize polyphony tend to pay a lot more attention to the development of a variety of scales, while emphasizing the nuances of tuning.
So -- even if we make the assumption that just because we don't have written polyphony in the Western tradition that it never occurred before the year 900 or so -- there's no good reason why an appreciation of interval relationships should lead to development of polyphony. Again, Western musical culture is very unusual compared to most traditional musics around the world in that regard.