Yes they do, but the primary reason bicycles (as well as motorbikes) have stability is because of trail - the contact point of the front wheel is behind the steering axis.
A banked-over bicycle naturally tends to turn into the corner because of trail, and that turning-in torques the bike back towards upright - turning the handlebars towards the centre of the curve moves the bottom of the bike towards the centre of the curve, and that pulls the bike upright.
The larger the trail, the more stable the bike is. Sportsbikes tend to have shorter trail so that they turn in faster, cruisers tend to have longer trails so they are more stable on long straight roads.
Loss of gyros would be far more noticeable at low speeds (less than 20mph) than high speeds IMO.
While trail and gyroscopic forces both influence steering and stability, neither are required. It has been shown that a bike can still be self-balancing even without trail or gyroscopic forces, though positive trail and gyroscopic forces do make it easier to balance and more stable (see http://bicycle.tudelft.nl/stablebicycle/StableBicyclev34Revi...).
ithkuil's comment is the more accurate description of what is going on.
ithkuil is saying the same thing as me, in a different way. I'm talking about how bikes are self-stable, he is talking about how bikes are stable with active input. Trail creates a steering feedback system. You need feedback from lean into the steering in order to create something self-stable, and trail is the simplest way of doing that.
http://en.wikipedia.org/wiki/Bicycle_and_motorcycle_geometry...
A banked-over bicycle naturally tends to turn into the corner because of trail, and that turning-in torques the bike back towards upright - turning the handlebars towards the centre of the curve moves the bottom of the bike towards the centre of the curve, and that pulls the bike upright.
The larger the trail, the more stable the bike is. Sportsbikes tend to have shorter trail so that they turn in faster, cruisers tend to have longer trails so they are more stable on long straight roads.
Loss of gyros would be far more noticeable at low speeds (less than 20mph) than high speeds IMO.