In fact, yes, they are! Even though "grapes" doesn't have the term "berry" in its name, it is still a true berry because the pericarp (fruit wall) is fleshy all the way through.
In botany, the berry is the most common type of simple fleshy fruit in which the entire ovary wall ripens into an edible pericarp. The flowers of these plants have a superior ovary and one or more carpels within a thin covering and fleshy interiors. The seeds are embedded in the common flesh of the ovary. Examples of botanical berries include the tomato, grape, lychee, loquat, lucuma, plantain, avocado, persimmon, eggplant, guava, uchuva (ground cherry), and chili pepper. Believe it or not, raspberries, blackberries, and strawberries are not true berries, but rather aggregate fruits—fruits that consist of multiple smaller fruits.
Hope I helped!