Phase-separated, circular monolayer domains at the air-water interface are unstable due to dipolar repulsions that tend to elongate the domain shapes. The corresponding monolayer motions are viscously coupled to the underlying liquid that acts to resist these shape changes. The flow field in a sublayer of finite depth is calculated and the growth rate of small domain shape disturbances is determined using linear stability analysis. The special case of a very shallow aqueous sublayer is considered, where the monolayer flow field is irrotational except close to the domain boundary. The calculations are applicable to fluid lipid monolayers.