I've met a few people who think 'classical' is music with strings and no vocal. Other schooled musicians will refer music composed within a certain period of time (1750-1824) as classical. Before that would be Baroque, after that Romantic and then Post-Romantic. All that is helpful for learning how styles influenced each other and how ideas evolved and developed.
Gershwin poses some interesting style points in his music. He came up through the pop music of his time, but his mind and ear were more open to possibilities than many others of his time. To call "Rhapsody in Blue" a classical piece is not academically correct. Also it is not really jazz. However it is art music in a pop (or early jazz) style.
Much the same kind of situation with "Sgt. Pepper's". You would sound foolish to call it 'Rock and Roll", and in a lot of ways, it's not Rock.... its not Progressive Rock. But it is art music in a rock/pop style.
A true genius will take the best from many ideas and create something new. Gershwin was such a genius, and its a damn shame that he died at age 37. His pop songs were as good as anyone else's from the 1930's, but his orchestral (art) music was pointing to something quite new. Check out "Porgy and Bess" and you'll hear music theater, jazz, and opera. Also worth a listen is "American In Paris".
Never confuse beauty with things that put your mind at ease. -Charles E. Ives