Haven't RTFA'd, but aren't libraries a public good? Yes, when I was a university student I barely used the physical library, relying instead on papers I could get online, etc., but the reason I could get access to those paper databases was because the school funded my access, and I paid for part of it with my tuition. These were academic databases - as far as I knew, there was no equivalent database of fiction, non-fiction, etc., just bookstores, Amazon.com et al., and, yes, libraries. At my local library in town, I've got a library card (that was either free or $2, can't remember which at present) that provides me access to an immense set of writing for gratis. What large-scale databases of books - that is, mainstream current literature, not things that have gone out of copyright, such as Project Gutenberg - are present? If some version of the Kindle is both cheap and lets users read current books both legally and for free (some kind of checkout system?), the library seems likely to remain.
On the point of public good: libraries are for everyone, not just the tech-savvy. Sure, it's great that people are getting connected, it's great that people have the disposable income to buy lots of books, but libraries serve many populations of people, not just those who are on the proper side of the Digital Divide. Do we really think that the digitally illiterate population of America (given that the topic is American, not that the digital divide is just a national issue) will have dropped to nothing by 2019? Because I am really dubious of that.
tl;dr Just because many people on one side of the digital divide have disposable income and like their books on terminal screens, it doesn't mean that a service provided for the good of the populace as a whole will whither and die.