This work analyzes the design and performance of a planar ionospheric gradient monitor, based on differential carrier-phase-only measurements between the elements of an antenna array. Small ionospheric disturbances are one of the potential threats to aerospace navigation, and have to be monitored to prevent an excessive degradation of the position estimates. A carrier-phase-based ionospheric gradient monitor enhances bias detection capabilities with respect to monitors that only exploit pseudorange measurements, provided that the ambiguous nature of the phase observations and the measurement error statistics are correctly handled. In this work we introduce a statistical test for detecting small ionospheric gradients, and we derive the analytical expression of the test probability distribution function for single- and two-baseline antenna arrays. A theoretical evaluation of the performance of the ionospheric gradient monitor is provided through inspection of the associated acceptance region, which defines the magnitudes of ionospheric gradient that cannot be detected with given integrity requirements. Due to the integer ambiguity of the carrier-phase measurements, multiple undesirable acceptance regions are produced. We provide a method to reduce the number and size of such regions.