Bear in mind the historical context. The chromatic scale wasn't invented until the 13th century. To the Greeks, scales were composed of a series of notes above (and sometimes below) a centrism, with various intervals. What's more, these intervals were neither exact half steps nor exact whole steps; the equally tempered scale didn't come until the 15th century (or later, depending who you wish to credit). Instead, the gaps between notes were determined by physics, in particular the lengths of strings or pipes in the ratios of whole numbers.
Even up through the 1800's, there was no single standard scale, and musicians and composers could invent different types of scales for different purposes, e.g. Kirnberger tuning. Even Bach's Well-Tempered Clavier may not have been intended for equal temperament-- the purpose of the preludes and fugues instead may have been meant to demonstrate the beauty of Bach's own tuning, where he tended to make thirds a bit sharp (citation) to allow for enharmonics in his compositions.
This allowed him to compose pieces that modulated in ways far more complex than his peers.
So what is a diatonic scale? Well, you can be derive one yourself very easily:
- Pick a note, and cut a string to length so that it will produce that note
- Take another string and cut it shorter than the first, in a ratio of exactly 2:3. This will be a perfect fifth above.
- Repeat step 2. This will be two fifths above, or a ninth. Now, this will be a bit high, so now double it in length so that it stays in the right octave.
- Keep cutting strings a fifth above/an octave below, working your way around the circle of fifths, until you reach a note that is just a bit higher than your first note (and sounds really bad).
- Stop right there.
If you follow this process, you end up with a scale that has seven notes. It happens to be in Lydian mode, but we can get the other modes through inversion, including the Ionian "major" mode/scale. You don't even need to think about half steps or whole steps. For convenience, you can name these notes with letters A through G. Presto, a diatonic scale!
Now this will sound very different from the modern, equally-tempered scale, which doesn't use the 2:3 ratio at all. Instead, it uses a ratio computed as the 12th root of 2 (~1.0594)-- the only way to get twelve half steps that are exactly equal, or to get the modern whole step. And the Greeks did not approach it that way at all.
So you see, if you look at the physics of music in use by the Greeks, the very notion of a half step or a whole step is non sequitur. It doesn't even fit into the equation. A diatonic scale is just something that has seven notes in it, with whole number ratios that yield a pleasant sound.