Tracking the resonance frequency of a mechanical oscillator is the basis of atomic force microscopy, mass sensing and thermometry. Nanomechanical thermal detectors exploit this principle to sense absorbed radiation, most effectively with tensile-stressed resonators, whose tension converts thermal expansion into a large frequency shift and dilutes dissipation. Approaching their fundamental sensitivity limit, however, has required free-space or fibre-coupled interferometers, hindering compact deployment and scaling to arrays. Here we remove this bottleneck by monolithically integrating a high-Q nanomechanical resonator with a silicon nitride photonic integrated circuit. A tensile-stressed silicon nitride trampoline with an impedance-matched platinum absorber is dispersively coupled to a photonic integrated microresonator beneath it. At room temperature and with microwatt-level optical power, the detector reaches a noise-equivalent power of 3.4 pW/√(rm Hz) for broadband thermal radiation (3-12 μm), referred to incident power. Referred to absorbed power, this is within a factor of 1.5 of its fundamental thermal fluctuation limit. Using our detector in a state-of-the-art commercial Fourier-transform infrared spectrometer, we resolve the vibrational fingerprint of a polymer film across 2-20 μm and reach the precision of the built-in pyroelectric detector with about an order of magnitude lower incident power. This architecture opens a route to multiplexed optomechanical focal-plane arrays.