No. The Gateway Arch in St. Louis is a solid stainless-steel structure with no opening large enough for an aircraft to pass through.
The Arch's sleek, aerodynamic shape and immense scale might suggest a fly-through is possible. However, the legs are solid up to the 300-foot level, and the interior space narrows dramatically toward the top. Even if there were an opening, FAA regulations and the Arch's status as a national monument would prohibit such a stunt.
The only things that "fly through" the Arch are the trams carrying visitors to the observation deck at the top.