Robotic ultrasound (US) calibration is essential for accurately relating US images to the robot coordinate system, but accurate and automated calibration remains challenging because existing methods often require complex phantoms or external 3D trackers. In this work, we develop a tracker-free robotic US calibration framework using a spherical-marker phantom and a highly automated perception pipeline for sphere-center localization. The proposed threshold-free image-processing method localizes the spherical feature based on each US image's intensity distribution, eliminating hand-tuned intensity thresholds and improving robustness across imaging settings without per-system retuning. Multi-pose observations of the spherical fiducial are then used to estimate the US-to-EE transformation without external tracking or prior localization of the sphere center in the robot base frame. We validate the proposed framework on a robotic US platform through repeated sphere scans and further assess the calibrated system using geometrically distinct phantoms with known CAD models. Across 12 cross-validation folds, single-marker calibration achieved sphere-center accuracy and precision of $1.75\pm0.51$ mm and $0.85\pm0.11$ mm, respectively, compared with $1.72\pm0.51$ mm and $0.84\pm0.12$ mm for three-marker calibration. Across the three reconstruction sets, the single-marker calibration yielded pooled post-registration point-to-surface MAE$\pm$SD values of $0.41\pm0.35$ mm for the cone and $0.50\pm0.41$ mm for the triangular prism, closely matching the three-marker results of $0.40\pm0.35$ mm and $0.48\pm0.40$ mm, respectively.

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