It's hardly integral for that though. They sponsor everything from spacejumps to skateboarding. They'd get by without it.
F1's viewership figures have fallen off the cliff since Bernie took pay TV's money. From over 600 million in 2009 to just over 500 million in 2011. It's a declining sport.
Re: viewership, you can always argue, but it's undeniable that a lot of the decrease is due to Bernie putting F1 on pay TV everywhere.
As for other sports, you are kidding yourself if think any other extreme sport RB sponsor has the same impact as F1. Shaun White is probably the only RB-sponsored sportsman I can name off the top of my head. And maybe people remember that dude who jumped from the stratosphere.
F1 has become RB's crown jewel which is now in danger because they are not dominating like they used to.
EDIT: They've also heavily invested in a grand prix, from sponsorship to actually owning the track and renovating it for 2010s' F1. That shows a lot of commitment from Red Bull; throwing it all away because things are not going their way seems rather stupid and short-sighted. Mateschitz is a smart man; he will calm down.