Fill a large pot with lightly salted water and bring to a rolling boil; stir in tortellini and return to a boil. Cook uncovered, stirring occasionally, until the tortellini float to the top and the filling is hot, about 3 minutes.

While the tortellini is cooking, whisk olive oil, balsamic vinegar, distilled white vinegar, Italian herb seasoning, salt, and pepper to make the dressing.

Drain tortellini, then rinse with cold water and drain again. Transfer to a large bowl and add artichoke hearts, mozzarella, pepperoni, olives, and green onions.