Preheat grill for medium heat and lightly oil the grate. Season chicken breasts with salt and black pepper.
Cook chicken breasts on the preheated grill until no longer pink in the center and the juices run clear, about 30 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Cut chicken into cubes.
Bring a large pot of lightly salted water to a boil; cook the rotini at a boil until tender yet firm to the bite, about 8 minutes; drain. Season pasta with garlic salt.
Whisk vegetable oil, lemon juice, basil, and Parmesan cheese together in a large bowl. Add pasta, chicken, and spinach leaves. Toss to coat.