Very impressive that he can whistle two notes at once anyway

I would say Träumerei by Schumann.

Edit: Confirmed

I played it out to see if I could tell, but I don't recognize it. Its not a particularly romantic or melodic thing tbh. Kinda mid.

I'm terrible about reading notational music, but... isn't that "Happy Birthday to you?"

Someone else already confirmed it, but here's a page with some trivia like that:

https://en.wikibooks.org/wiki/Annotations_of_The_Complete_Peanuts/Print_version

It's kind of hit-or-miss, but does list this one:

Charlie Brown is delivering the traditional whistle of appreciation for feminine beauty (though it usually has two notes: WEEEET-WOOO), and Patty takes offense. Schroeder, being more musical, delivers a mini-concert to Violet and gets to walk off with her. It is the melody to "Traumerei" from Schumann's "Scenes from Childhood" for piano.