What's wrong with my Persian lime leaves, and what should I do about it?
This looks like the effect of some kind of mite. Usually washing the plant with soap and water takes care of the problem. If not, horticultural oil may work.
Other possibilities are citrus leaf miner and aphids. I don't see leaf miner tunneling, so I don't think this is it, but look closely. Aphids would be visible on the underside of the leaves, but you said you didn't see any insects.
And only being indoors does not prevent these various insects from feasting on your plant.
When using insecticides, always read the label. Here is a reference: