Purpose: To develop and evaluate DosimeTron, an agentic AI system for automated patient-specific MC internal radiation dosimetry in PET/CT examinations. Materials and Methods: In this retrospective study, DosimeTron was evaluated on a publicly available PSMA-PET/CT dataset comprising 597 studies from 378 male patients acquired on three scanner models (18-F, n = 369; 68-Ga, n = 228). The system uses GPT-5.2 as its reasoning engine and 23 tools exposed via four Model Context Protocol servers, automating DICOM metadata extraction, image preprocessing, MC simulation, organ segmentation, and dosimetric reporting through natural-language interaction. Agentic performance was assessed using diverse prompt templates spanning single-turn instructions of varying specificity and multi-turn conversational exchanges, monitored via OpenTelemetry traces. Dosimetric accuracy was validated against OpenDose3D across 114 cases and 22 organs using Pearson's r, Lin's concordance correlation coefficient (CCC), and Bland-Altman analysis. Results: Across all prompt templates and all runs, no execution failures, pipeline errors, or hallucinated outputs were observed. Pearson's r ranged from 0.965 to 1.000 (median 0.997; all p < 0.001) and CCC from 0.963 to 1.000 (median 0.996). Mean absolute percentage difference was below 5% for 19 of 22 organs (median 2.5%). Total per-study processing time (SD) was 32.3 (6.0) minutes. Conclusion: DosimeTron autonomously executed complex dosimetry pipelines across diverse prompt configurations and achieved high dosimetric agreement with OpenDose3D at clinically acceptable processing times, demonstrating the feasibility of agentic AI for patient-specific Monte Carlo dosimetry in PET/CT.