I think when it comes to screen time, it's important to differentiate between passive and active screen time. Passive screen time is when the person is in "receive only" mode (e.g., watching a movie, a YouTube video). Active screen time is when the person is engaged (e.g., playing a video game, using an app, on a Skype call with friends). There are studies that indicate that the brain processes these two types of screen time very differently.
I agree with polarbear that mixing it up and providing a variety of experiences, both in front of the screen and away from it, is beneficial. And I think that following a child's lead by subject matter (gifted or not) makes everyone's lives easier. As long as you temper that with health and well being in mind... so maybe don't feed ice cream to your kid for dinner on a regular basis.
