Medical ultrasound has been widely used to examine vascular structure in modern clinical practice. However, traditional ultrasound examination often faces challenges related to inter- and intra-operator variation. The robotic ultrasound system (RUSS) appears as a potential solution for such challenges because of its superiority in stability and reproducibility. Given the complex anatomy of human vasculature, multiple vessels often appear in ultrasound images, or a single vessel bifurcates into branches, complicating the examination process. To tackle this challenge, this work presents a gaze-guided RUSS for vascular applications. A gaze tracker captures the eye movements of the operator. The extracted gaze signal guides the RUSS to follow the correct vessel when it bifurcates. Additionally, a gaze-guided segmentation network is proposed to enhance segmentation robustness by exploiting gaze information. However, gaze signals are often noisy, requiring interpretation to accurately discern the operator's true intentions. To this end, this study proposes a stabilization module to process raw gaze data. The inferred attention heatmap is utilized as a region proposal to aid segmentation and serve as a trigger signal when the operator needs to adjust the scanning target, such as when a bifurcation appears. To ensure appropriate contact between the probe and surface during scanning, an automatic ultrasound confidence-based orientation correction method is developed. In experiments, we demonstrated the efficiency of the proposed gaze-guided segmentation pipeline by comparing it with other methods. Besides, the performance of the proposed gaze-guided RUSS was also validated as a whole on a realistic arm phantom with an uneven surface.