The solution is what it always was. Design in units like 'em's, don't override the user defaults, and let the page scale however the end user wants. This not only accomodates the retina/non-retina split, it lets users with bad eyesight increase the font size and still be able to read things.
Images? That's a bit harder, but it's not the end of the world. Vector images are useful here, but for higher detail raster images, designing in double size and scaling down for ordinary screens gets you a long way. For smaller icons, it might pay to have two versions, and I don't see a way around that for a while. Gnome has handled this for a long time by having specific sizes of icon that it tried to snap to, as well as vector fallbacks for when none of the sizes were appropriate.
The important thing is to give up on pixel perfection. Just let the user chose the size, and don't mess with their defaults.
I don't think designers have to give up on "pixel perfection" - well, they don't have to give everything up.
We already have an enormous gamut of displays for any website. Forcing iOS users to a broken "mobile website" is never going to satisfy the designers who care about being pixel perfect, so they were grappling with these issues already.
And I can't say it enough: we're finally getting past 1920x1080 screens again. CRTs had higher resolutions a decade ago.
Some of us _do_ have good enough eyesight, and some of us _do_ work with videos that big (and the associated images) - yes, there's the Apple 30" Cinema Display, but I seek real competition between manufacturers.
I agree, designing in pixels has always been the wrong way to go about it. It is only needed when there is a restriction in the number of pixels available to work with(such as displays that have large pixels.)
The beauty of high pixel density screens is that we can literally output our vectors/high resolution rasters to the output size and not have to worry about massaging individual pixels for the best clarity. The advent of high pixel density screens allows us to exclusively think in terms of the end size on screen, instead of being bogged down with pixels dimensions. This high density makes design easier, not harder. Also the concept of different pixel densities is not new, screens have always had different pixel densities.
The reliable thing about 'retina' screens is that we can think of the pixel problem as 'solved' and just prepare artwork to pass the retina test, instead of trying to match it perfectly for every higher pixel density out there. (A level of accuracy that won't be easily seen by the user.) The same thing is done in print everyday, 300 dpi, 600 dpi, 800 dpi, it doesn't matter, past a certain point the end user isn't going to casually notice the extra detail.
Designers that have been working in print would see the analogy to various print device resolutions which are each measured in lines per inch. Again the approach is to think in terms of the final dimensions, and not get bogged down with the individual device resolutions.
I understand that designing for 1x can be problematic, but it's the same workflow as designing for any foreign pixel ratio/pixel density (such as non square pixels used in certain types of film.) It's just another step in the work flow and testing often on a device is useful way for the designer gets the hang of it.
Images? That's a bit harder, but it's not the end of the world. Vector images are useful here, but for higher detail raster images, designing in double size and scaling down for ordinary screens gets you a long way. For smaller icons, it might pay to have two versions, and I don't see a way around that for a while. Gnome has handled this for a long time by having specific sizes of icon that it tried to snap to, as well as vector fallbacks for when none of the sizes were appropriate.
The important thing is to give up on pixel perfection. Just let the user chose the size, and don't mess with their defaults.