The most common malignancies in the world are skin cancers, with melanomas being the most lethal. The emergence of Convolutional Neural Networks (CNNs) has provided a highly compelling method for medical diagnosis. This research therefore conducts transfer learning with grid search based hyper-parameter fine-tuning using six state-of-the-art CNN models for the classification of benign nevus and malignant melanomas, with the models then being exported, implemented, and tested on a proof-of-concept Android application. Evaluated using Dermofit Image Library and PH2 skin lesion data sets, the empirical results indicate that the ResNeXt50 model achieves the highest accuracy rate with fast execution time, and a relatively small model size. It compares favourably with other related methods for melanoma diagnosis reported in the literature.