This is a very interesting question, but probably quite hard to answer. I'll weigh in, but with the hopes that an opthalmologist or someone expert in ancient medicine might come along later.

It's quite likely Greek doctors in the Roman period would have known that floaters were some kind of impurity within the vitreous humour of the eye, but I can't find any reference to it off-hand. Aetiologically, they may well have explained it in terms of an imbalance of the eye's natural properties (hot/cold and wet/dry).

Here is Galen (Method of Medicine, 1.49K) on the lens of the eye:

"A part (morion) is a homoiomere if it is divisible in its entirety into like parts, as the name itself clearly indicates, just as, in the eye, the transparent vitreous and crystalline, and the specific substance of the tunics (cornea, sclera) are. [...]. The organ is the eye while sight is the function. One of its parts is homoiomerous and is the primary organ of sight, which is the crystalline humor (lens), as has been demonstrated in my works on these, since this is what needs to be altered by one of the colors externally for the animal to see. However, it cannot be altered unless it is perfectly clear and transparent, and it cannot be clear and transparent like this apart from the mixture that now constitutes it. For it was shown that each existing thing is as it is because of the mixture of hot, cold, moist, and dry, so that, if any of these [qualities] is altered greatly, the animal will either not see, or will see badly."