(1) Background: Previous evidence has indicated a connection between a Mediterranean diet and cardiovascular disease. However, evidence for subclinical markers of cardiovascular disease, such as arterial stiffness, is limited. Therefore, the aim of this study was to assess the associations between adherence to the Mediterranean diet (MD), as assessed by the MEDAS-14 questionnaire, and arterial stiffness, as assessed by aortic pulse wave velocity, in healthy adults and according to sex. (2) A cross-sectional study including 386 healthy participants was performed in the EVasCu study. Adjusted and unadjusted differences in adherence to the MD and arterial stiffness were determined using Student's t test and ANCOVA for the total sample and according to sex. (3) Results: Our results showed that individuals with a high adherence to the MD had a greater arterial stiffness, both in the total sample and in females, although this difference was not significant after adjusting for possible confounding variables, such as age. (4) Conclusions: Our findings indicated that, in the unadjusted analyses, healthy subjects with a high adherence to the MD showed a greater arterial stiffness. When these analyses were adjusted, no significant differences were shown in a-PWv according to the categories of MD adherence.