Could a plane be constructed to be fly in fixed-stick roll-stable circles?
This will become another of my long posts; after all, lateral stability is more complex than longitudinal stability and involves many more factors. Short answer: It is possible, but not by aerodynamic means, and would incur a drag penalty.

Starting condition

Assume we have an aircraft with fixed controls, trimmed for straight flight, and make it a glider to remove propulsion effects. Next, let it fly through an asymmetric gust which lifts one wing. For what follows we assume calm air again.

Initial sideslip

For an outside, earth-fixed observer, a component of lift is pointing sideways and not compensated by weight, so the aircraft will accelerate to that side. From the point of view of the aircraft, lift is still acting in the plane of symmetry, but gravity does not and will cause it to sideslip. This will mainly evoke these reactions:


Directional stability $c_{n\beta}$: The aircraft will yaw into the wind because the vertical tail will create a side force ("weathervane effect").
Dihedral effect $c_{l\beta}$: The additional lift on the windward wing will lift this wing. Also, the additional side force on the vertical tail, which yaws the aircraft, will cause a supporting rolling moment. Roll damping ($c_{lp}$) will limit the amount by which the aircraft rights itself up, however.
Yaw moment due to dihedral and wing sweep: The lift difference due to dihedral and wing sweep will retard the upwind wing, causing a yawing motion into the wind. This effect can be added to the directional stability $c_{n\beta}$.


Eventual (almost) coordinated turn

The aircraft will start to fly a coordinated turn, because all forces try to drive sideslip to almost zero, and on their way to do so start a yawing motion. In contrast to that, the roll angle will not change further once sideslip stops, because the sideslip-induced rolling moments stop as well. However, if a roll angle remains, the aircraft will now begin to turn. This opens up a new bunch of effects, because  now we have asymmetric flow: Due to the yawing motion, airspeed varies over wingspan and sideslip angle varies over length:


Rolling moment due to yawing motion $c_{lr}$: The outer wing (which was the leeward wing during sideslip before) now travels faster, creating more lift. This causes a rolling moment which increases roll angle.
Since the angle of attack would be the same in a coordinated turn over the whole wingspan, drag will behave proportional to lift and increase with the local turn radius. This drag difference will create a small yawing moment which will let the inner wing advance, until the dihedral effect will kick in and create enough of a difference in angle of attack that the drag on the inner and on the outer wing are equal again. Since induced drag is proportional to the square of the lift coefficient, the wing will still create less lift on the inner wing than on the outer wing, but the difference will be reduced. If the airplane climbs or descends during the turn, angle of attack will vary over span and will create an uprighting moment on the descending airplane (and vice versa).
Yawing moment due to yawing motion $c_{nr}$: This is also called yaw damping and creates a yawing moment opposed to the yawing motion. Contributing factors are the vertical tail which sees a sideward flow component due to the yawing motion, and the drag distribution along the wingspan. Aircraft with a long tail have high yaw damping and, in combination with plenty of dihedral, tend to be more roll stable than short-tailed aircraft. Here the long tail forces the wing into a sideslip which uprights the aircraft. This configuration is found on free-flying models which do indeed upright themselves from shallow banks.
Centrifugal forces: The sideward component of lift is now balanced by the centrifugal force due to the turn. However, the centrifugal forces also will create an uprighting rolling moment which depends on the spanwise distribution of masses.




Note that the outer wing has a bigger turn radius $R$, but the same angular velocity $\omega$ as the rest of the airplane. This causes it to experience more centrifugal force $m\cdot \omega^2\cdot R$ than the inner wing (symbolized by the length of the parallel arrows). I used a rather steep attitude to get the point across, but the same holds true at shallower bank angles. This is the uprighting inertial moment.

Depending on the relative size of these effects, the aircraft will either right itself up, stay at this roll angle or dive ever deeper into the turn. Agile aircraft with low roll inertia will have too little uprighting moment and are likely to spiral dive. More stable configurations with big vertical tails and ample of dihedral will either continue to turn gently or right themselves up.

Effects at non-zero roll rates

Now this was all done under the assumption that the roll angle changes very slowly. A too small vertical tail and/or too much dihedral will excite the Dutch roll motion, and now the roll angle oscillates, adding more roll-induced effects. To keep this answer reasonably short, I will not list them here.

Sweptback configurations optimized for fast flight (= with small vertical tails close to the wing) are typical examples where the ratio $\dfrac{c_{l\beta}}{c_{n\beta}}$ is too high and the Dutch roll eigenmode has too little damping. Free flying model aircraft, on the other hand, cannot afford to fall into a spiral dive, and by giving them big vertical tails and long lever arms, which allows them to have substantial dihedral, and a high radius of roll inertia, they can be made to upright themselves.