Nomophobia refers to the contemporary fear of being unable to communicate sufficiently via a smartphone. As reported in the literature, nomophobia leads to excessive smartphone use, and one of the crucial issues of this overuse is its effect on physical and mental health. The current study aimed to investigate the association between perceived health assessments and nomophobia among young adult smartphone users through the mediating role of depression and stress. A cross-sectional study was conducted on 1408 young adults aged 18–25 in Athens, the capital city of Greece. Data were collected through a self-reported questionnaire and included information on sociodemographic characteristics, patterns of smartphone use, self-perceived health status, and depressive and stress symptoms. Nomophobia was assessed using the “Nomophobia Questionnaire (NMP-Q)”. The effect of nomophobia on perceived health was analyzed by taking into account its effect on depression and stress. The results indicate a positive significant association between nomophobia and overall health burdening (i.e., musculoskeletal, hearing/vision, and psychosomatic symptoms). Depression and stress seem to play a significant mediating role in this association. Raising awareness through health-promoting interventions could play a pivotal role in eliminating the phenomenon of nomophobia and its consequences.