Human eye contact is a form of non-verbal communication and can have a great influence on social behavior. Since the location and size of the eye contact targets vary across different videos, learning a generic video-independent eye contact detector is still a challenging task. In this work, we address the task of one-way eye contact detection for videos in the wild. Our goal is to build a unified model that can identify when a person is looking at his gaze targets in an arbitrary input video. Considering that this requires time-series relative eye movement information, we propose to formulate the task as a temporal segmentation. Due to the scarcity of labeled training data, we further propose a gaze target discovery method to generate pseudo-labels for unlabeled videos, which allows us to train a generic eye contact segmentation model in an unsupervised way using in-the-wild videos. To evaluate our proposed approach, we manually annotated a test dataset consisting of 52 videos of human conversations. Experimental results show that our eye contact segmentation model outperforms the previous video-dependent eye contact detector and can achieve 71.88% framewise accuracy on our annotated test set. Our code and evaluation dataset are available at https://github.com/ut-vision/Video-Independent-ECS.