(1) Background: The present study aimed to investigate the onset of mental disorders in the six months following hospitalization for COVID-19 in people without a previous psychiatric history. (2) Methods: This was a longitudinal study carried out among adults who had been hospitalized due to COVID-19 infection. Six months after discharge, a series of questionnaires were administered (the World Health Organization Well-being Index (WHO-5), the Patient Health Questionnaire-9, the General Anxiety Disorder Questionnaire-7, and the Drug Abuse Screen Test, among others). Based on these scores, a compound Yes/No variable that indicated the presence of common mental disorders was calculated. A multivariate logistic regression was built to explore the factors associated with the presence of common mental disorders. (3) Results: One hundred and sixty-eight patients (57.34%) developed a common mental disorder in the 6 months following hospital discharge after COVID-19 infection. Three variables were independently associated with the presence of common mental disorders after hospitalization for COVID-19, and the WHO-5 duration of hospitalization), and severity of illness. (4) Conclusions: Among people with no previous psychiatric history, we observed a high incidence of mental disorders after COVID-19 hospitalization. A moderate (1–2 weeks) duration of hospitalization may pose a higher risk of post-COVID-19 onset of a mental health condition than longer or shorter durations of medical hospitalization. Further research is needed to better understand the mechanisms underlying the psychopathological consequences of COVID-19 and their predictors.