The invention discloses a reinforced concrete arch bridge construction method, which comprises the following steps of: 1, constructing a pile foundation, a pile cap, a bridge abutment body side wall and a bridge pier body; 2, erecting arch ring supports from the fourth hole to the eighth hole and from the thirteenth hole to the seventeenth hole on two sides of an arch bridge, arranging a locking system, and finally pouring concrete to arch rings of the eight holes; 3, detaching the arch ring supports in the step 2, erecting arch ring supports from the zero hole to the fourth hole and from theseventeenth hole to the twenty-first hole on the bridge abutment side, arranging a locking system and finally pouring concrete to arch rings of the eight holes; and 4, detaching the arch ring supports erected in the step 3, erecting arch ring supports from the eighth hole to the thirteenth hole in the span, arranging a locking system, pouring concrete to arch rings of the five holes, and folding the whole bridge. Main bodies of the arch ring supports have arch truss structures, a characteristic that steel bars are tensioned is utilized, the structures are firm, the middles of the structures are not required to be supported, materials are saved, the arch ring supports are convenient to detach, and constriction time is saved.