Heat oil in a nonstick skillet over medium-high heat. Add onion; cook, stirring occasionally, until tender, about 2 minutes. Stir in spinach; cover and cook until wilted, about 1 ½ minutes. Remove from heat; stir in marinara and chicken.
Preheat an air fryer to 325 degrees F (165 degrees C) according to manufacturer's instructions.
Divide dough into 4 equal pieces. Roll each piece on a lightly floured work surface into a 6-inch circle. Spread ¼ of the spinach mixture over the bottom half of each dough circle. Top each with ¼ of the mozzarella cheese. Fold dough over filling to form half-moons and crimp edges to seal. Coat well with cooking spray.
Transfer calzone to the air fryer basket. Cook for 8 minutes. Turn calzone over; cook until golden brown, about 4 minutes more.
