Why would my microwave continually rotate when connected to a voltage stabilizer which is also used for a refrigerator?

You should / MUST not connect stabilizer made for refrigerator to microwave oven. Stabilizer for fridge will be of capacity around 0.5kw and that for microwave oven should be around 2kw or more.

As already said, it could be due to surges at the gate firing the Microwave triac that controls the power to the rotation motor. These firing could be caused by non-sinusoidal voltages coming from the stabilizer.

A power overload is unlikely since the motor needs less then 100W. And the powerfull magnetron should not run. Is it a combination MW with grill and hot air?

Normally the door switches will completely switch off a MW, at least the magnetron. As suggested by other posters, a simple comparison at a normal outlet with doors opened/closed could help to find the problem.