I did some more checking around today. I was able to turn up two references to silk being spun from
Saturnia pyri's cocoons,
here and
here. The first link states that a "high quality silk" has been made from the cocoons, while the second states that silk has been spun, but "not commercially".
This link states that the fibers of Greek moths would produce a rougher silk, but it names three species in three genera and doesn't specify which one makes this "rougher" silk... or if it's referring to all of them.
This similarly just mentions that silk from this species "has been established".
This link tells us that the fibers are intermediate in quality between common silk and wild silks from East Asia.
So, we know that it's at least possible, but there's a couple of problems with it just looking at the photos. For one, the threads are kind of dark. Now, apparently feeding the caterpillars different things and different points in their lifecycle can affect the color, so I'll have to do some more reading here on what is available and what colors it would produce, but we also might try developing a means of bleaching that doesn't damage the cloth, perhaps a recipe similar to what the Dutch came up with during the Medieval Period. That wouldn't even have to be original to the silk weavers, though. I see no reason why silk can't be made under a strict trade secret in one place and then shipped somewhere else for bleaching. The Dutch were masters of bleaching for centuries because of their secret recipe, or so I have read. For two though, the problem of the shortness of the threads seems apparent in some, but not all of the photos I've seen of the cocoons. Might this be something that's taken care of over time through selective breeding? I was looking at some photos of
cocoons used for tussah silk, and they don't look entirely that much more sophisticated.
Even if we can't get something as high quality as what's coming out of China, if the Iberians can monopolize the market in the Mediterranean by the Late Bronze Age, does it really matter? I don't see it really mattering, at that point. There will be an established, multi-layered market with a lot of fingers in the pie by the time Chinese start making contact with the Middle East. I think that Chinese silk would eventually win out over Iberian silk in the long run, but that process might be very slow, but the takeover I think would be slow, especially in Western Europe, where silk would be traded along the Atlantic Coast.