The development of low-power wearable systems requires specialized techniques to accommodate their unique requirements and constraints. While significant advancements have been made in the inference phase of artificial intelligence, the training phase remains a challenge, particularly for biomedical wearable systems. Traditional training algorithms might not be suitable for these applications due to the substantial memory requirements and high computational costs associated with processing the large number of bits involved in neural network operations. In this paper, we introduce a novel learning procedure specifically designed for low-power wearable systems, dubbed Bio-BPfree (deep neural network training without backpropagation for low-power wearable systems). Using a two-class classification task, Bio-BPfree replaces conventional forward and backward backpropagation passes with four forward passes, two for data of the positive class and two for data of the negative class. Each layer is equipped with a unique objective function aimed at minimizing the distance between data points within the same class while maximizing the distance between data points from different classes. Our experimental results, which were obtained by conducting rigorous evaluations on the MIT-BIH dataset that features electrocardiogram (ECG) signals, effectively demonstrate the superior performance and suitability of Bio-BPfree for two-class classification tasks, particularly within the challenging environment of low-power wearable systems designed for continuous health monitoring and assessment.