The COVID-19 pandemic and pandemic-induced lockdowns and quarantine establishments have inevitably affected individuals, businesses, and governments. At the same time, the spread of the COVID-19 pandemic had a dramatic impact on financial markets all over the world and caused an increased level of uncertainty; the stock markets were no exception either. Most of the studies on the impact of the COVID-19 pandemic on stock markets are based either on the analysis of a relatively short period (the beginning of pandemic) or a longer period, which, in turn, is very heterogeneous in terms of both the information available on the COVID-19 virus and the measures taken to contain the virus and address the consequences of the pandemic. However, it is very important to assess the impact not only at the beginning of the pandemic but also in the subsequent periods and to compare the nature of this impact; the studies of this type are still fragmentary. Therefore, this research aims to investigate the impact of the COVID-19 pandemic on stock markets of two of the most severely affected European countries—Italy and Spain. To reach the aim of the research OLS regression models, heteroscedasticity-corrected models, GARCH (1,1) models, and VAR-based impulse response functions are employed. The results reveal that the stock market reaction to the spread of the COVID-19 pandemic differs depending on the country and period analyzed: OLS regression and heteroscedasticity-corrected models have not revealed the statistically significant impact of the spread of the COVID-19 pandemic, while impulse response functions demonstrated the non-zero primary response of analyzed markets to the COVID-19 shock, and GARCH models (in the case of Spain) confirmed that the COVID-19 pandemic increased the volatility of stock market return. This research contributes to the literature by providing a comprehensive impact assessment both during the whole pre-vaccination period of the pandemic and during different stages of this period.