Absolutely you can.
The verb do is extremely flexible, and is often used in the context of makeup and grooming.
A woman might say she's doing her eyes, or doing her cheeks, or doing her hair. In that context, do essentially means taking care of.
It seems the verb do is more likely to be used in this way when there's more than one step involved. For example, if a woman is simply applying lipstick, she might be more inclined to say:
I'm putting on lipstick.
but if she's planning to use a lip pencil, lipstick, and lip gloss, then she might be more inclined to say:
I'm doing my lips.
Because the eyes are often made up with several steps, "I'm doing my eyes" is a very normal thing to say.
That said, @Inazuma makes a good point. The more flexible a verb is, the greater the chance someone might think it means something else. If someone said, "I'm doing my eyes tomorrow," that could mean she was planning to use mascara and eye shadow, but, conceivably, it could mean she was scheduled to have cataract surgery in morning. So long as the context makes it clear what you mean, though, doing my eyes is a perfectly acceptable substitution for the lengthier doing my eye makeup.