Measurements on single biomolecules provide a perspective of exceptional detail, enabling understanding of the underlying molecular mechanisms that govern biological processes. Most commonly, single-molecule measurements employ fluorescence which often requires the chemical attachment of fluorescent labels. As well as being perturbative to biomolecules and their interactions, labels limit the achievable resolution and sensitivity through complex photophysics. Recent developments in single-molecule label-free Rayleigh scattering based approaches have enabled the determination of the molecular weight of single biomolecules down to 30 kDa, however these techniques require proximity to surfaces. Microscale optical fiber based Fabry-Pérot cavities (FFPCs) provide a fully integrated route towards high sensitivity, label-free single-molecule measurements in the solution-phase. In this work we exploit both the small mode volume and high Q-factor of high reflectivity biconcave FFPCs to measure the properties of single proteins down to 1 kDa as they undergo Brownian motion in aqueous solution with signal-to-noise-ratios of >100. Here we improve the limit of detection 30-fold and the temporal resolution 100-fold compared to the current state-of-the-art. With this label-free technique we are able to determine the hydrodynamic radii of single proteins with unprecedented sensitivity. This technology has the potential to be developed into a highly sensitive diagnostic tool for complex diseases such as Alzheimer's and Parkinson's which require detection of small rare protein species. This has substantial implications for medical sciences and the ability to develop drugs and therapies for complex diseases.