A correlated colour temperature tunable, four-component, LED system suitable for illuminating traditional Chinese paintings painted with inorganic pigments (iop-TCPs) has been designed. This light source can meet the requirements for protective illumination and colour quality in museum illumination. The damage to iop-TCPs caused by the individual 450 nm, 510 nm, 583 nm and 650 nm monochromatic LEDs used to construct the LED system were obtained through a long-term illumination experiment. Then, we calculated the damage resulting from the corresponding four-component white LEDs with various spectral power distributions, the intensities of which were iterated by a brute force algorithm. The constructed spectral power distributions were evaluated by the damage and colour quality formulae. The results showed that the better colour quality spectral power distributions with higher correlated colour temperatures cause less damage to iop-TCPs. The lowest damage spectral power distributions, satisfying colour quality requirements with correlated colour temperatures ranging from 2700 K to 4000 K, were obtained by calculating the corresponding parameters of the formed spectral power distributions and further selection. This paper not only provides a colour temperature tunable, four-component, white LED system suitable for illuminating iop-TCPs but also sets out a methodology that can be used to identify white LEDs suitable for other cultural relics based on damage limitation and visual requirements.