Melanoma is one of the deadliest forms of skin cancer, but early and accurate identification can significantly improve the survival rate of patients. In this paper, an end-to-end framework based on multi-instance learning is proposed for melanoma recognition and lesion segmentation simultaneously. To make full use from the information of high-resolution images, we take each image block (super-pixel) as an instance in a bag and use multi-instance learning based on a graph convolutional network to recognize melanoma. Moreover, skin lesion segmentation is derived from attention weights and is calibrated by classification probability vectors. As a result, the AUC of our method for melanoma recognition reaches 0.93, which is much higher compared with other related methods. Also, the Jaccard index (JA) of our method for melanoma-related skin lesion segmentation reaches 0.699. In our end-to-end approach, segmentation and recognition are treated as intimately coupled processes, and hence, a high JA is also an indication of the reliability of melanoma recognition. Collectively, these findings confirmed that our method effectively assists melanoma diagnosis.