Non-contact ultrasonic thickness measurement is important for industrial metrology, particularly for metallic components where surface temperature, motion, geometry, or contamination can limit conventional contact probes. In this study, we present a fully non-contact thickness measurement system for aluminum plates integrating laser-ultrasound excitation, 3×3 interferometric detection, and deep learning-based phase demodulation. Broadband ultrasonic waves were generated using a pulsed 532 nm laser in the ablation regime, while rear-surface displacement was detected using a 1550 nm fiber-optic interferometric probe. Two phase-shifted outputs from a 3×3 coupler were processed by a neural interferometric demodulator (NID), which directly reconstructed continuous optical phase without separate quadrature reconstruction, arctangent phase extraction, or phase unwrapping during inference. PZT-modulated measurements at 10, 15, 20, and 25 kPa provided additional validation against conventional 3×3 standard ellipse fitting (3×3-SEF), including phase excursions exceeding $2\mathsf{\pi }$. The NID reduced the phase RMSE from 0.048 to 0.071 rad to 0.029–0.041 rad and the processing time from 481 ± 22 μs to 295 ± 30 μs per sequence. For thickness measurement, the reconstructed phase was converted into out-of-plane displacement, and acoustic time-of-flight (TOF) features were extracted. For 2, 6, and 10 mm aluminum plates, the estimated thicknesses were 1.911 ± 0.024 mm, 6.102 ± 0.070 mm, and 10.516 mm with a partial standard uncertainty of 0.127 mm, respectively. These results demonstrate accurate and efficient deep learning-assisted phase demodulation for fully non-contact laser-ultrasonic thickness gauging.