It won't run DX, and OgL isn't up to scratch from a developer point of view and titles already out won't get an overhaul, so no.
This is what I'm hoping The PS4 does, PC spec hardware that is very similar to what will be available shortly and in the future from AMD specifically but likely Intel in not too long as well. But they are basically AFAIK running linux and opengl. Also importantly AMD were exclusively supporting windows OS, while they had linux drivers they weren't actively pushing for their vendors to sell anything but windows PC's. They've dropped that a few months ago and well DX for windows is no where near what it should be.
Keep in mind MS get to control it, not the devs, where as openGL can be if more supported and pushed hard(seemingly on PS4, a potential Valve box and more serious support for Linux from AMD themselves(and as such they need something other than DX support on it).
In doing so PS4/AMD/dev's can break MS's monopoly on gaming which is great for everyone but MS and fantastic for gamers. Buy a AMD pc without the extra £100 tacked on for an MS OS... so PC's just got £100 cheaper, and full support of openGL as a standard that will hopefully be sensibly optimised where as Window's DX has many problems. Effectively ending up with a faster PC with the same hardware AND saving the cash on the OS...
AS far as I know Sony use their own version of openGL because they need lower level access than Windows would allow. THe two things that are possible is for AMD to get together with some linux guys and allow low level access and work on a unified version. But Sony were using a modified version largely because their hardware in the PS3 was pretty significantly different to desktop, where as now the hardware will be vastly closer.
Either way they are effectively optimising all their game code for openGL in one form or another and if AMD and Valve threw their entire weight behind openGL pushing into the future, it's an advantage for Sony to work with them to improve openGL as an API because even if they don't directly use it(opengl "normal" was available on ps3 but rarely used because more power was available through the lower level api, and opengl will be available on ps4 as well) improvements to openGL can surely be translated and implemented in their own api updates.
Ultimately that is three HUGE companies involved in a huge amount of gaming who all would very much like to break MS's monopoly of DX and/or to gain a performance advantage.
AMD currently could well do with it, think about £400 laptop vs an Intel laptop where the CPU costs £50-100 more, when you take out the OS tax of £50-100, the difference between an AMD/Intel laptop becomes a much larger percentage and also pushes AMD laptop prices down towards tablet pricing.
One of the HUGE, absolutely huge disadvantages all MS OS based laptops, tablets/mobile devices have vs anything with Android is they are automatically significantly more expensive. Think Surface RT being more expensive than a Android tablet and that is with MS discounting the OS, all their partners basically dumped their versions of RT notebooks because the OS cost made them instantly uncompetitive with android laptops and with the Surface as they were charging themselves less for the OS on Surface than any of their partners. So if say Asus built an idential spec to Surface it would cost more than it due to the OS, it's strangling sales of the Surface as it's too expensive and destroyed sales of other devices with Window's RT. AMD's biggest chance to capitalise on cheap "good enough" cpu's in mobile devices is to drop the huge OS tax and that means dropping MS and supporting anything not OS... meaning Valve, Sony, openGL/Linux.