Pretty much what MiracleDinner said. I don't think there's any long-term medical conditions from staring at a screen, but it does cause eyestrain and excessive exposure to blue light can interfere with your ability to sleep, as well as secondary effects from being stuck in a computer for too long such as a strained wrist from using the mouse too much. You should take breaks if you feel like you're staring too much, get up and walk around for a bit.
Think about it this way: a huge amount of people work in a screen as part of their jobs, such as game developers. If there were documented cases of long-term health consequences (mind you, purely from staring at a screen, ignoring other confounding variables such as long hours and crunch periods), the game development jobs would have one of the most extensively documented cases of it, don't you think?