Egocentric videos provide a first-person perspective of the wearer's activities, involving simultaneous interactions with multiple objects. In this work, we propose the task of weakly-supervised Narration-based Video Object Segmentation (NVOS). Given an egocentric video clip and a narration of the wearer's activities, our aim is to segment object instances mentioned in the narration, without using any spatial annotations during training. Existing weakly-supervised video object grounding methods typically yield bounding boxes for referred objects. In contrast, we propose ROSA, a weakly-supervised pixel-level grounding framework learning alignments between referred objects and segmentation mask proposals. Our model harnesses vision-language models pre-trained on image-text pairs to embed region masks and object phrases. During training, we combine (a) a video-narration contrastive loss that implicitly supervises the alignment between regions and phrases, and (b) a region-phrase contrastive loss based on inferred latent alignments. To address the lack of annotated NVOS datasets in egocentric videos, we create a new evaluation benchmark, VISOR-NVOS, leveraging existing annotations of segmentation masks from VISOR alongside 12k newly-collected, object-based video clip narrations. Our approach achieves state-of-the-art zero-shot pixel-level grounding performance compared to strong baselines under similar supervision. Additionally, we demonstrate generalization capabilities for zero-shot video object grounding on YouCook2, a third-person instructional video dataset.