An electronic device may be provided with wireless circuitry that includes a phased antenna array. The array may include multiple antennas each having multiple antenna feeds for covering different polarizations. Control circuitry may control the wireless circuitry to transmit signals at millimeter or centimeter wave frequencies using a first set of feeds in the array and at least one selected phase. The wireless circuitry may receive the signals transmitted by the first set of feeds using a second set of feeds in the array. The control circuitry may gather phase measurements for the received signals and may compare the phase measurements to the selected phase to generate phase difference values. The control circuitry may perform external object proximity detection operations based on the phase difference values. The control circuitry may control the wireless circuitry to cycle through different combinations of antenna feeds for the first and second sets.