Bring a large pot of lightly salted water to a boil. Cook zoodles in the boiling water, stirring occasionally until tender yet firm to the bite, about 10 minutes. Scoop out some of the cooking water and reserve. Drain zoodles.
Heat oil in a large skillet over medium heat. Add chicken, garlic and pepper; cook and stir until fragrant, 1 to 2 minutes. Add zoodles and Pecorino Romano cheese. Ladle in 1/2 cup of reserved cooking water; stir until cheese is melted, about 1 minute. Add more cooking water until sauce coats spaghetti, about 1 minute more.