An electronic device may be provided with control circuitry and wireless circuitry. The wireless circuitry may include a phased antenna array and a radio-frequency integrated circuit having transmit and receive ports. The array may include a first set of stacked patch antennas coupled to the transmit ports and a second set of stacked patch antennas coupled to the receive ports. The integrated circuit may transmit ranging signals at millimeter wave frequencies using the transmit ports and the first set of antennas. The integrated circuit may receive a reflected version of the transmitted ranging signals that has been reflected off of an external object using the receive ports and the second set of antennas. The control circuitry may identify a distance between the electronic device and the external object based on the transmitted and received signals.