We propose, implement, and evaluate a method for nowcasting the daily number of new COVID-19 hospitalizations, at the level of individual US states, based on de-identified, aggregated medical insurance claims data. Our analysis proceeds under a hypothetical scenario in which, during the Delta wave, states only report data on the first day of each month, and on this day, report COVID-19 hospitalization counts for each day in the previous month. In this hypothetical scenario (just as in reality), medical insurance claims data continues to be available daily. At the beginning of each month, we train a regression model, using all data available thus far, to predict hospitalization counts from medical insurance claims. We then use this model to nowcast the (unseen) values of COVID-19 hospitalization counts from medical insurance claims, at each day in the following month. Our analysis uses properly-versioned data, which would have been available in real-time, at the time predictions are produced. In spite of the difficulties inherent to real-time estimation (e.g., latency and backfill) and the complex dynamics behind COVID-19 hospitalizations themselves, we find overall that medical insurance claims can be an accurate predictor of hospitalization reports, with mean absolute errors typically around 0.4 hospitalizations per 100,000 people, i.e., proportion of variance explained around 75%. Perhaps more importantly, we find that nowcasts made using medical insurance claims can qualitatively capture the dynamics (upswings and downswings) of hospitalization waves, which are key features that inform public health decision-making.