We develop an adjustment-free differential-phase demodulator based on a delay-interferometer (DI) made of a 3 x 3 optical coupler, which is used as a 120-degree optical hybrid, and demonstrate the possibility of using it as a phasor monitor for the multilevel differential phase-shift keyed (DxPSK) signal. The key features of the proposed demodulator are twofold: (a) in-phase and quadrature (I-Q) components and the phasor diagram can be obtained only-by using a single DI, and (b) the phase delay of the DI can be derived from the output power of the DI without the knowledge of the signal under test. These features enable us to demodulate the I-Q components of DxPSK signals without any adjustments. We formulate a theoretical model for deriving the differential phasor from the output signals of the 3 x 3 coupler regardless of unbalances in the phase retardations and the splitting ratios of the 3 x 3 coupler. Thus, the discrepancy from the ideal 120-degree optical hybrid is not a problem, and hence the requirements of the 3 x 3 couplers are greatly relaxed. For a demonstration, we implement the proposed demodulator by using a fiber Michelson interferometer with Faraday rotator mirrors and use it as a differential phasor monitor capable of the plug-and-play (wavelength- and polarization-independent) operation. We show that this phasor monitor can be used for diagnosis of differential phase-shift keyed (DPSK) and differential quadrature phase-shift keyed (DQPSK) transmitters by identifying the sources of impairments from the measured constellation diagram and phasor trajectories. The proposed phasor monitor can also be used for monitoring the optical signal-to-noise ratio (OSNR) of DPSK signal.