X-ray examination is suitable for screening of gastric cancer. Compared to endoscopy, which can only be performed by doctors, X-ray imaging can also be performed by radiographers, and thus, can treat more patients. However, the diagnostic accuracy of gastric radiographs is as low as 85%. To address this problem, highly accurate and quantitative automated diagnosis using machine learning needs to be performed. This paper proposes a diagnostic support method for detecting gastric cancer sites from X-ray images with high accuracy. The two new technical proposal of the method are (1) stochastic functional gastric image augmentation (sfGAIA), and (2) hard boundary box training (HBBT). The former is a probabilistic enhancement of gastric folds in X-ray images based on medical knowledge, whereas the latter is a recursive retraining technique to reduce false positives. We use 4,724 gastric radiographs of 145 patients in clinical practice and evaluate the cancer detection performance of the method in a patient-based five-group cross-validation. The proposed sfGAIA and HBBT significantly enhance the performance of the EfficientDet-D7 network by 5.9% in terms of the F1-score, and our screening method reaches a practical screening capability for gastric cancer (F1: 57.8%, recall: 90.2%, precision: 42.5%).