Developing Text-to-Speech (TTS) systems that can synthesize natural breath is essential for human-like voice agents but requires extensive manual annotation of breath positions in training data. To this end, we propose a self-training method for training a breath detection model that can automatically detect breath positions in speech. Our method trains the model using a large speech corpus and involves: 1) annotation of limited breath sounds utilizing a rule-based approach, and 2) iterative augmentation of these annotations through pseudo-labeling based on the model's predictions. Our detection model employs Conformer blocks with down-/up-sampling layers, enabling accurate frame-wise breath detection. We investigate its effectiveness in multi-speaker TTS using text transcripts with detected breath marks. The results indicate that using our proposed model for breath detection and breath mark insertion synthesizes breath-contained speech more naturally than a baseline model.