The "damage" done by screens is in the form of eye strain. Your eyeball focuses by using muscles in the eye to change the shape of the lens within the eye, to properly focus incoming light onto the correct portion of the retina.
It is the second part of your eye, after the cornea, that helps to focus light and images on your retina. Because the lens is flexible and elastic, it can change its curved shape to focus on objects and people that are either nearby or at a distance.
VisionAware: Anatomy of the Eye
Since we are talking about muscles physically manipulating the lens, constantly focusing on items at the same location will cause the actual muscles to fatigue. This manifests itself as:
Eye strain also known as asthenopia is an eye condition that manifests itself through nonspecific symptoms such as fatigue, pain in or around the eyes, blurred vision, headache, and occasional double vision. Symptoms often occur after reading, computer work, or other close activities that involve tedious visual tasks.
Wikipedia: Eye Strain
Once the muscles have had a chance to rest, the body recovers. Since eye-strain is a short-term condition, you are not actually "damaging" the eye.
Eye strain facts. ... Extended computer use or inadequate or excessive lighting may cause eye strain, but there are no permanent consequences of this. Symptoms can include headaches, blurring of the vision, feelings of dryness, and other discomfort, but eye strain will not damage your eyes or change their anatomy.
Medicinenet.com: Eye Strain
Eyestrain doesn't have serious or long-term consequences, but it can be aggravating and unpleasant.
(Wish I could down-vote Mayo Clinic for using "aggravating" on their web site like that).
Mayo Clinic: Eyestrain complications
As one ages, like with many other types of tissues in the body, the lens of the eye becomes less flexible/more rigid, making it more difficult to manipulate for focus. This would speed the onset of and increase the frequency of eye strain, as the muscles have to work harder. This is why people (like me, who avoided glasses for almost 50 years) need reading glasses more often as they age.