Retinopathy of prematurity (ROP) is a sight threatening disorder that primarily affects preterm infants. It is the major reason for lifelong vision impairment and childhood blindness. Digital fundus images of preterm infants obtained from a Retcam Ophthalmic Imaging Device are typically used for ROP screening. ROP is often accompanied by Plus disease that is characterized by high levels of arteriolar tortuosity and venous dilation. The recent diagnostic procedures view the prevalence of Plus disease as a factor of prognostic significance in determining its stage, progress and severity. Our aim is to develop a diagnostic method, which can distinguish images of retinas with ROP from healthy ones and that can be interpreted by medical experts. We investigate the quantification of retinal blood vessel tortuosity via a novel U-COSFIRE (Combination Of Shifted Filter Responses) filter and propose a computer-aided diagnosis tool for automated ROP detection. The proposed methodology involves segmentation of retinal blood vessels using a set of B-COSFIRE filters with different scales followed by the detection of tortuous vessels in the obtained vessel map by means of U-COSFIRE filters. We also compare our proposed technique with an angle-based diagnostic method that utilizes the magnitude and orientation responses of the multi-scale B-COSFIRE filters. We carried out experiments on a new data set of 289 infant retinal images (89 with ROP and 200 healthy) that we collected from the programme in India called KIDROP (Karnataka Internet Assisted Diagnosis of Retinopathy of Prematurity). We used 10 images (5 with ROP and 5 healthy) for learning the parameters of our methodology and the remaining 279 images (84 with ROP and 195 healthy) for performance evaluation. We achieved sensitivity and specificity equal to 0.98 and 0.97, respectively, computed on the 279 test images. The obtained results and its explainable character demonstrate the effectiveness of the proposed approach to assist medical experts.