This paper introduces MicroClinic, an ultra-compact convolutional neural network designed for medical image analysis under extreme resource constraints. While state-of-the-art architectures typically rely on millions of parameters, MicroClinic operates in a regime of 0.4k to 1.1k trainable parameters, following a design philosophy where relational processing substitutes parameter redundancy. The architecture integrates lightweight convolutional blocks with a Convolutional Multi-Head Attention (CMHA) mechanism to capture global spatial dependencies without increasing network depth. Benchmarked across twelve independent clinical datasets, MicroClinic achieves competitive performance, reaching 99.9% accuracy on MedicalMNIST and 95.7% on COVID-19 X-Ray classification, effectively matching models up to 29,000 times larger in parameter count. Beyond efficiency, the reduced scale limits memorization (attain zero error) capacity, potentially improving data privacy, while also enabling structural analysis of the learned representations through metrics such as the Fisher discriminant ratio and mutual information. These results demonstrate that diagnostically meaningful accuracy can be achieved within a minimal parameter budget, enabling AI-assisted screening in underserved and latency-sensitive clinical environments or on hardware with limited computational resources.