Early detection of highly infectious respiratory diseases, such as COVID-19, can help curb their transmission. Consequently, there is demand for easy-to-use population-based screening tools, such as mobile health applications. Here, we describe a proof-of-concept development of a machine learning classifier for the prediction of a symptomatic respiratory disease, such as COVID-19, using smartphone-collected vital sign measurements. The Fenland App study followed 2199 UK participants that provided measurements of blood oxygen saturation, body temperature, and resting heart rate. Total of 77 positive and 6339 negative SARS-CoV-2 PCR tests were recorded. An optimal classifier to identify these positive cases was selected using an automated hyperparameter optimisation. The optimised model achieved an ROC AUC of 0.695 ± 0.045. The data collection window for determining each participant's vital sign baseline was increased from 4 to 8 or 12 weeks with no significant difference in model performance (F(2) = 0.80, p = 0.472). We demonstrate that 4 weeks of intermittently collected vital sign measurements could be used to predict SARS-CoV-2 PCR positivity, with applicability to other diseases causing similar vital sign changes. This is the first example of an accessible, smartphone-based remote monitoring tool deployable in a public health setting to screen for potential infections.