This paper proposes a neuromorphic processing system and its classifier design for always-on wearable ECG classification. The ECG signal is captured by LC sampling yielding single-bit temporal coding that can be natively fed into an SNN in an event-driven manner. Such an architecture simplifies the quantization of ADC and bypasses the coding processing for SNN. Thus, the system power can be reduced by simplified data conversion architecture, single-bit data representation for input data reduction, and spare processing of SNN. Spatiotemporal backpropagation training is optimized to adapt to the LC-based data representation and mitigate the firing rate, thus increase network sparsity. The system-level design of the hardware architecture consisting of an LC-ADC and an SNN processor is evaluated by Simulink-ModelSim co-simulation. Trained with the MIT-BIH database, the proposed system achieves 95.34% in classification accuracy with an average of 79 sampling points and 24.6 kFLOPs per inference, corresponding to 55.9x and 42.4x reduction on sampling data and FLOPs per inference respectively, compared with conventional ADC and ANN approaches.