Comment by 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
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.