I would have thought that an eye doctor would mention that there are two types of cells that detect light in the human eye. Cone cells are responsible for seeing colour, and work if there is plenty of light, while rod cells are more sensitive to low light intensity and are responsible for night vision and more importantly peripheral vision. When noticing something in fine detail we are using the cone cells. When it is too dark in the room we are getting the information from the rod cells, but our vision will not be as sharp.
If you are reading in the middle of the night, with no lights turned on in the room, your pupil might try to adjust in such a way as to see something in the periphery (rod cells), meaning that there will be too much light coming to your cone cells. Some people might not rely much on peripheral vision, and therefore the pupil will adjust to the screen, and not cause any problems.
And this is true also during the day. I don't have problems reading on an LCD screen, and my vision is still good (I'm 27), but I tried reading something printed on a perfectly white paper, outside, when the sun was shining, and unless I sat in the shade, the light reflected of the piece of paper would hurt my eyes. My night vision is great by the way, I usually move around the house at night with only the street light that comes through the window for guidance.