In a large pot, heat the olive oil over medium-high heat. Add the celery, onions, and carrots and cook, stirring occasionally, until the vegetables begin to soften, about 3 to 4 minutes.
Add the chicken broth to the pot; bring to a boil.
Once the broth has come to a boil, gently add the meatballs, one at a time, then give them a gentle stir to make sure they don't stick together.
Add the acini de pepe; stir for a few seconds to prevent the pasta from sticking together.
Cover partially with a lid and cook at a low boil for 20 minutes or until the pasta is tender and the meatballs are cooked through (internal temperature of at least 160°F for beef, pork, veal and lamb; 165°F for turkey or chicken).
Turn off the heat. Taste and adjust seasoning if needed.
Add the spinach; stir until wilted.