While traditional deep learning algorithms have achieved impressive performance for automatic arrhythmia detection, such methods typically require significant computational resources, making them infeasible for continuous on-device monitoring. In this work we introduce a neuromorphic architecture, a hierarchically gated recurrent spiking neural network (HG-RSNN). Our network augments a typical recurrent spiking neural network (RSNN) with learnable gates and top-down connectivity. Learnable gates are used to restrict feature sharing between layers and across time, allowing for the extraction of multi-timescale features. Visualization of gate activity highlights important regions of ECG waveforms, providing a method for understanding information flow through the model. We evaluate model performance on a binary arrhythmia detection task using ECG recordings acquired from the PhysioNet MIT-BIH Arrhythmia Database. Our model achieves average area under the receiver-operator curve of 0.902 showing an improvement of +0.120 over a standard recurrent spiking neural network. We further assess model performance when trained using a variant of eligibility propagation (e-prop). e-prop is a biologically plausible and memory efficient approximation of the backpropagation-through-time algorithm (BPTT), the standard algorithm for training recurrent neural networks. e-prop condenses past information into constant-memory eligibility traces, reducing memory requirements. We show that our model can be effectively trained using e-prop. We see marginal loss in performance compared to BPTT, illustrating how our method could be used for direct on-device learning.