The identification and segmentation of histological regions of interest can provide significant support to pathologists in their diagnostic tasks. However, segmentation methods are constrained by the difficulty in obtaining pixel-level annotations, which are tedious and expensive to collect for whole-slide images (WSI). Though several methods have been developed to exploit image-level weak-supervision for WSI classification, the task of segmentation using WSI-level labels has received very little attention. The research in this direction typically require additional supervision beyond image labels, which are difficult to obtain in real-world practice. In this study, we propose WholeSIGHT, a weakly-supervised method that can simultaneously segment and classify WSIs of arbitrary shapes and sizes. Formally, WholeSIGHT first constructs a tissue-graph representation of WSI, where the nodes and edges depict tissue regions and their interactions, respectively. During training, a graph classification head classifies the WSI and produces node-level pseudo-labels via post-hoc feature attribution. These pseudo-labels are then used to train a node classification head for WSI segmentation. During testing, both heads simultaneously render segmentation and class prediction for an input WSI. We evaluate the performance of WholeSIGHT on three public prostate cancer WSI datasets. Our method achieves state-of-the-art weakly-supervised segmentation performance on all datasets while resulting in better or comparable classification with respect to state-of-the-art weakly-supervised WSI classification methods. Additionally, we assess the generalization capability of our method in terms of segmentation and classification performance, uncertainty estimation, and model calibration. Our code is available at: https://github.com/histocartography/wholesight.