Purpose: Smartphone addiction is prevalent among medical students, and there is a concern that the coronavirus disease 2019 (COVID-19) pandemic fueled a rise in smartphone addiction. Earlier studies suggest a link between excessive smartphone usage and negative outcomes such as depression, stress, and reduced academic achievement. However, there is a dearth of both local studies in Saudi Arabia and studies conducted during the COVID-19 pandemic exploring the prevalence of smartphone addiction and its association with academic performance, depression, and perceived stress, which is the purpose of the current study. Methods: In 2021, a cross-sectional research project took place among medical students at King Saud University and the Vision Colleges located in Riyadh, Saudi Arabia. An online self-administered questionnaire consisting of demographic variables, grade point average (GPA), the Patient Health Questionnaire-9 (PHQ-9), the Perceived Stress Scale-4 (PSS-4), and the Smartphone Addiction Scale—Short Version (SAS-SV) was deployed. Results: Three hundred and fifteen students participated. Around 47.9% of students reported smartphone addiction, and the mean SAS-SV score was 32.31 ± 12.01 points. Both PHQ-9 and PSS-4 scores showed a significant positive correlation with the SAS-SV score (r = 0.216, p < 0.001 and r = 247, p < 0.001, respectively), while GPA did not (r = −0.027, p = 0.639). An adjusted analysis showed that the PSS-4 score was positively associated with the SAS-SV score (odds ratio (OR) = 1.206, p < 0.001), while the PHQ-9 score was not (OR = 102, p = 0.285). Conclusions: Smartphone addiction is prevalent among medical students and associated with perceived stress. Additional research is required to gain a deeper comprehension of this issue and to assess the success of intervention initiatives aimed at encouraging healthy smartphone usage, particularly in times of crisis like the COVID-19 pandemic.