Recent studies have found a relationship between fear of COVID-19 and mental health problems. Medical workers caring for COVID-19 patients tend to suffer from mental health problems; however, the impact of their personality traits, in the form of mental problems like depression and anxiety in Japan is unclear. In this study, we investigated the risk of nurses’ depression and anxiety, predicted by the fear of COVID-19 and the Big Five personality traits. A total of 417 nurses working in hospitals providing care to COVID-19 patients in Wakayama prefecture of the Kansai region participated in this study. The questionnaires comprised items on nurses’ basic characteristics and three scales: the Fear of COVID-19 Scale 2020, the Big-Five Scale, and the Japanese version of the Hospital Anxiety and Depression Scale (HADS). Depression and anxiety in the HADS were set as dependent variables, and basic attributes, fear, and personality traits as independent variables; multivariate logistic regression analyses were conducted. The questionnaire, with no missing items was distributed from February to March 2021. Neuroticism (OR = 1.06, 95%CI = 1.03–1.09) was the only significant factor associated with the depression symptom, and both FCV-19S scores (OR = 1.16, 95%CI = 1.09–1.23) and neuroticism (OR = 1.09, 95%CI = 1.06–1.13) were the significant factors associated with anxiety. The Nagelkerke’s R squared was 0.171 in the depression model and 0.366 in the anxiety model. Thus, it was found that it is necessary to support nurses’ mental health by developing methods suitable to their personalities.