That's not the point.
The point is... Well, there are multiple points.
First, performance. A proper OS like Windows 7/8 or OS X is pretty heavy on performance. Windows can easily use somewhere around 1.5 GB or RAM - without anything running. Just the OS being there will require a decent bit of the hardware to be dedicated to it. Consoles do run their own operating systems, sure. But if you compare Sony's XMB or Microsofts Xbox Dashboard to a full blown operating system, you'll notice that the consoles' OS are far more lightweight (and limited) so they don't have to handle the load of an actual operating system. That's the advantage of it being a console going right out of the window, which was what I mentioned earlier.
Second, price. Licensing something like Windows or OS X would be pretty pricey. Linux isn't very mainstream and would, probably, put off more users than it would attract due to not being developed for the mass market in the same way as Windows and Mac OS are. Developing a operating that's comparable of either of these would cost a decent bit of mone in the first place - and you'd still have a huge problem with compatibility (which would be why you'd be going for a proper OS in the first place). And I'd assume that neither Apple, who are selling their own hardware, nor Microsoft who's a direct competitor to Sony would be happy to license their OS to Sony.
Third, not a closed system. That's right, you can do pretty much anything you want with a full-blown operating system like Windows. You wouldn't have to jail-break a PS4 with stuff like that. With ever increasing DRM, how likely is Sony to actually lower the amount of DRM they can enforce on the player by giving them an open operation system to work with? Unlikely, to say the least.
What you might see is Linux support, sure. But who on earth cares about Linux? The PS4 isn't going to do crap in the PC market if it comes with Linux. Just as a little reminder, Windows accounts for more than 91% of the market share of operating systems in the computer market. Mac OS has slightly more than 7%. Linux doesn't even account for 1.5%. Unless the PS4 has either Windows or Mac OS, it won't be a competitor to the PC market. And, as alluded to above, MS will probably to whatever it can to prevent Windows from ever running on a PS4. Apple too, given that the PS4 would cut into their very own market share in that case.
So, what OS would Sony put on it? Linux isn't going to do crap and the only major players are competitors.
Huge thread to the PC market, yeah, sure. What's so darn hard about thinking this through, logically, for once? Sure, you might get Linux, but nobody will really care. And that might be the case if Sony thinks it is worth the expense to open the console up for it - which might or might not happen. They even halted the support on the PS3 specifically after it was implemented, which doesn't indicate a huge desire to support an operating system that's entirely insignificant, anyways.
Can we please stop with the "OMG PS4 gunna stomp PC market so hard" stuff? It's not a PC and it will not do well in the PC market unless it runs one of the established operating system, which doesn't include Linux. So, if you guys could stop your wishful thinking for a bit, that'd be great.