The PlayStation2 is based on a MIPS chip, 128bit VLIW EmotionEngine is a marvel of RISC design. It runs a proprietary 'system'. Developers write games on a Linux box with a cross compiler to target the MIPS. I.e. RFP-Real Fu*&ing Programming, none of this WinCE crap. The PS2 itself does not run Linux.
That said, you know darn well that Sony is gonna build a gaming network to rival AOL, and it will have a browser, etc. This is the real reason M$ 'announces' X-box, to try and deflect Sony. If PS2 matches PS with over 70million units worldwide, that about 3 times the AOL numbers.
Sony wants to pump video/audio on demand and with cable/dsl it might work. Imagine DVD quality movie Pay-per-View for pennies, without leaving your TV. Imagine the targeted advertising revenues. Imagine you're Bill Gates.
Did I mention you can plug up a SVGA monitor to a PS2 for highres pc style browsing? And the cool games for less $$ than PC games, well PS games are generally cheaper.