Integrating robotically driven contact-based material characterization techniques into self-driving laboratories can enhance measurement quality, reliability, and throughput. While deep learning models support robust autonomy, current methods lack reliable pixel-precision positioning and require extensive labeled data. To overcome these challenges, we propose an approach for building self-supervised autonomy into contact-based robotic systems that teach the robot to follow domain expert measurement principles at high-throughputs. Firstly, we design a vision-based, self-supervised convolutional neural network (CNN) architecture that uses differentiable image priors to optimize domain-specific objectives, refining the pixel precision of predicted robot contact poses by 20.0% relative to existing approaches. Secondly, we design a reliable graph-based planner for generating distance-minimizing paths to accelerate the robot measurement throughput and decrease planning variance by 6x. We demonstrate the performance of this approach by autonomously driving a 4-degree-of-freedom robotic probe for 24 hours to characterize semiconductor photoconductivity at 3,025 uniquely predicted poses across a gradient of drop-casted perovskite film compositions, achieving throughputs over 125 measurements per hour. Spatially mapping photoconductivity onto each drop-casted film reveals compositional trends and regions of inhomogeneity, valuable for identifying manufacturing process defects. With this self-supervised CNN-driven robotic system, we enable high-precision and reliable automation of contact-based characterization techniques at high throughputs, thereby allowing the measurement of previously inaccessible yet important semiconductor properties for self-driving laboratories.