I don't read "books" any more, not even e-books. But yes, I mostly ... or only read on screen. The truth is, it does not have any noticeable negative impact on my eyes. I have been wearing glasses since grade 3, and my eyesight has not deteriorated much in the last 10 - 15 years. I actually find the backlit screen easier to read than ink on paper.
More importantly, the Internet has changed my way of "reading". I now feed on more up-to-the-minute info worldwide, from news, magazine type articles, to academic journals (which require subscription). We now no longer have to wait for the published version of journals (which are 3 - 6 months late), not to say published books (which are usually more than 12 - 18 months late).