Monolithic integration of InSb on Si could be a key enabler for future electronic and optoelectronic applications. In this work, we report the fabrication of InSb metal-semiconductor-metal photodetectors directly on Si using a CMOS-compatible process known as rapid melt growth. Fourier transform spectroscopy demonstrates a spectrally resolved photocurrent peak from a single crystalline InSb nanostructure with dimensions of 500 nm x 1.1 mu m x 120 nm. Time-dependent optical characterization of a device under 1550 nm illumination indicated a stable photoresponse with responsivity of 0.50 A W-1 at 16 nW illumination, with a time constant in the range of milliseconds. Electron backscatter diffraction spectroscopy revealed that the single crystalline InSb nanostructures contain occasional twin defects and crystal lattice twist around the growth axis, in addition to residual strain, possibly causing the observation of a low-energy tail in the detector response extending the photosensitivity out to 10 mu m wavelengths (0.12 eV) at 77 K.