Given the independent nature of racing teams, and plethora of top level venues available, starting a competitor for F1 would be far simpler than creating a competitor for most other top level sports.
Naturally I assume this is some sort of bargaining tactic. I'm not quite sure what Bernie gets out of it. Maybe a consulting fee. Maybe his foot in the door as a way of gaining more power/money. Maybe he just misses 'the game' and is doing it for free. However, if anyone was going to create a competitor to F1, it would be Bernie. Even though I'm dismissing the idea out of hand... I wouldn't be surprised if he could have a competitive series up and going in 18 months. There's plenty of tracks available. A tweak to the rules allowing for customer chassis to start. I don't even know if F1 rules forbid the manufacturers from supplying identical engines to a second series.