Background: COVID-19 during pregnancy has been associated with various obstetric complications; however, epidemiological evidence regarding its relationship with hypertensive disorders of pregnancy (HDP) and preterm birth remains inconclusive. Therefore, we aimed to evaluate the association between COVID-19 during pregnancy and the risk of HDP and preterm birth among Mexican pregnant women. Methods: We conducted a retrospective cohort study based on a review of 3710 medical records of women treated at a tertiary hospital between 2020 and 2023. COVID-19 was diagnosed by real-time reverse transcription polymerase chain reaction (RT-PCR) assay. We estimated adjusted risk ratios (aRRs) using robust Poisson regression models, controlling for potential confounders. Results: The COVID-19 rate was 2.5%. We observed that women with COVID-19 had a higher risk of HDP (aRR = 1.59; 95% CI = 1.07–2.36). When HDP subgroups were analyzed separately, COVID-19 was associated with an increased risk of preeclampsia (aRR = 2.04; 95% CI = 1.34–4.20) and preterm birth (aRR = 1.50; 95% CI = 1.02–2.19). The association with gestational hypertension lost statistical significance after adjustment. Conclusions: Our findings suggest that COVID-19 during pregnancy may be associated with hypertensive disorders of pregnancy and preterm birth. However, the observed association with preeclampsia should be interpreted cautiously because diagnostic overlap between severe COVID-19 and preeclampsia cannot be excluded.