Why don't I have a custom title by now?!
No current Linux version can do that, for reasons that others have already cited but I'll gladly repeat: poor cross-distro binary compatibility and library dependency hell, which makes end-user-machine-specific compilation from source the preferred, if not the only practical way of distributing software, which in turn is incompatible with the way commercial videogames have been marketed until nowadays.
Their response? Step one, make Linux a supported Steam platform. Start porting Valve games to it, and encouraging other vendors to do the same.
Windows and Mac OS software was so easy to sell on a disk because everything comes statically linked and precompiled, on OSes with a predictable API, ABI, and any required libraries are either statically linked or provided during game installation from disk. In particular, game developers develops against those libraries (DirectX, Direct3D, PhysX etc.)
The problem with the Linux world? The statically-linked, precompiled executable is pretty much excommunicated like the Devil from most of the Linux community. No sane company would bother maintaining a dozen of slightly different executables in order to tackle distro- or window-manager specific bugs, and those that tried didn't exactly strike it rich.
This would work, but it would essentially create their own version of the "walled garden" model which triggered (hypothetically) this whole thing to begin with. So Steam would be the demons, and become a zombie themselves ;-) That is, unless they were far more liberal about licensing it and had a team of developers making sure that it works exactly the same on the various possible future hardware platforms (ARM is expected to gain momentum, though Intel will still be King, either in the old IBM PC-compatible BIOS flavor, or the newfangled UEFI or even MacIntel).
Step two, produce a fully integrated version of Linux that combines a stripped down version of the OS with Steam at a deep level. In essence, a Steam OS, which is given away free to hardware vendors, who pre-install it on commodity PC hardware to produce what are essentially Steam consoles.
For desktop, the "Steam OS" could presumably run in a VM of sorts adding yet another layer of isolation and abstraction, and yet another performance hit in the name of "enhanced end consumer services".