The biggest nationwide COVID-19 pandemic lockdown worldwide was enforced in India for an initial period of 21-days. Emerging evidence suggests that pandemic situations and associated lockdowns have an adverse impact on sleep and mental health. However, prediction of sleep health from sociodemographic characteristics and the public's worry during the initial stages of the COVID-19 pandemic has not been extensively explored so far. It's also unclear whether sleep outcomes mediate the association between worry and mental health during pandemic situations. A web-survey (N = 391) on sociodemographic characteristics, COVID-19 related worry, sleep health (insomnia and daytime sleepiness), and depression was conducted during the initial 21-days of the COVID-19 stringent lockdown in India. Multiple regression analyses showed that variables, including sex, age, income level, and worry score, contributed to the significant regression equation for insomnia but not for daytime sleepiness. Specifically, the female, younger, lower income, and highly worried populations contributed significantly more than the male, elderly, higher income, and less worried populations, respectively, to the prediction of insomnia. Mediation analyses showed that insomnia, but not daytime sleepiness, fully mediated the relationship between worry score and severity of depressive symptoms. We provide evidence that the female, younger, lower income, and worried populations may be at higher risk for insomnia during pandemic situations. Current evidence gives hope that improving sleep may reduce depressive symptoms during a pandemic situation. This underscores the importance of the implementation of effective public health policies in conjunction with strategical responses to the COVID-19 pandemic.