Hawking is wrong, and right for the wrong reasons and I'm surprised he can't see it (being a bit of a cyborg himself).
Humans will "evolve" into AI machines, obviously not evolve in the traditional sense, but we will be their ancestors - again not in the traditional sense (this is of course just the plot of AI where the future intelligence looks back fondly at the humans that have passed, as we look back fondly on our early epitherian ancestors), we humans that have detached ourselves from traditional evolution with all that pesky social care and medicine, which I think is a good thing (and I suspect that Hawking would agree).
So what does that mean for the future? well, us crappy meat sacks need food, water, air, we hate radiation and break randomly, but the AI of the future will be able to travel the galaxy, not bothered (as much) by getting old, only really, really old, they can just be backed up and restored to some newer shiny kit, replicate themselves, drawing resources from the planets, nuclear power that lasts for centuries, solar where available, the occasional asteroid etc. as it travels, shutting down for the odd millennia as it travels between the stars, carrying the knowledge of the world with it.
None of this is new, it's deeply rooted in the Sci-Fi we all know and love, from Star Trek with Voyager, through Independence Day and the Silver Surfer, the ideas are there, not just in Sci-Fi of course, the Von Neumann self replicator has 70 years of history, and perhaps is an argument why we could be unique (or at least the first) in our galaxy to get to this level of development - i.e. why hasn't another life form filled the galaxy with such machines?
What is sure, humans living off this biosphere has some serious problems, not just biodiversity, but basics like growing food and recycling all the products, perfect water harvesting, energy production (we need a lot), and strangely enough, if we solve all these problems, living on earth could be a lot better for those who don't have so much of that.