[bityard]

Sure you can.

At very low speeds, yes, the bike will fall over. But a bike with some minimum amount of speed can roll upright on its own just fine. You can try it yourself with your least favorite bike and an empty parking lot. All it takes is a good solid push. It has to do with bicycle frame geometry and center of mass.

Neil deGrasse Tyson explains it in this video at 2:52: https://www.youtube.com/watch?v=RIrgwWQqqts

[Gh0stRAT]

That's not countering the argument that steering is what is preventing the bike from falling over rather than the gyroscopic effect of the wheels. You'd have to tie off the handlebars with a static line before rolling it in order to prove that it was the gyroscopic effect keeping the bike upright.

[ajmurmann]

I can ride my bike easily without touching the handlebars at all once it's rolling.

[merlynkline]

Try that with the handlebars tied immovable. Or actually, please don't.

[powersnail]

I actually tried a it a lot when I was young, and it never worked even on a downward slope. Maybe I was too weak back then, but again, the point is that a full bicycle doesn't have the same behavior as a standalone wheel, which is very easy to roll, and pretty much stands on its own until its speed is really really low.