Social distancing is widely acknowledged as an effective public health policy combating the novel coronavirus. But extreme forms of social distancing like isolation and quarantine have costs and it is not clear how much social distancing is needed to achieve public health effects. In this article, we develop a design-based framework to test the causal null hypothesis and make inference about the dose-response relationship between reduction in social mobility and COVID-19 related public health outcomes. We first discuss how to embed observational data with a time-independent, continuous treatment dose into an approximate randomized experiment, and develop a randomization-based procedure that tests if a structured dose-response relationship fits the data. We then generalize the design and testing procedure to accommodate a time-dependent treatment dose in a longitudinal setting. Finally, we apply the proposed design and testing procedures to investigate the effect of social distancing during the phased reopening in the United States on public health outcomes using data compiled from sources including Unacast, the United States Census Bureau, and the County Health Rankings and Roadmaps Program. We rejected a primary analysis null hypothesis that stated the social distancing from April 27, 2020, to June 28, 2020, had no effect on the COVID-19-related death toll from June 29, 2020, to August 2, 2020 (p-value < 0.001), and found that it took more reduction in mobility to prevent exponential growth in case numbers for non-rural counties compared to rural counties.