I'd assume.. no.
There's almost no way either console does standard multi-tasking like that. Would be obnoxious.
Would you mind trying?
It might freeze and suspend the browser, like you say - I just figured it might work - since it works great doing the same with media-player on the vita, and I think the memory-budget for the browser is in the OS-section, not game-section, like Vita's media-player.
Havn't tried doing the same on Vita tough, just did the media-player stuff there.
On Vita some games automatically mute the soundtrack, and others you'll have to manually mute soundtrack in options. Sound effects play over, unless you mute them aswell.