If so, what would it sound like?
In the sense of rough scale invariance and self-similarity, sure, but it wouldn’t be absolute. For that matter you can’t really see the fractal nature of something like the Mandelbrot set – just an approximation.
Now, given that the real definition of a fractal has to do with a notion of “filling space”, what “dimensions” of what “space” does music fill to begin with? In that sense, the question doesn’t even parse at all.
Of course, maybe you mean "something that sounds (rather than looks) sorta cool and trippy and you’d find reproduced endlessly in college dorm rooms, in which case I’d say it’d sound like “The Wall”.
Google “fractal music.” There’s a lot of free software out there that does just this.
Or read Douglas Adams’ The Long Dark Tea-Time of the Soul. There’s nothing factual in it, and the idea of fractal music isn’t a significant part of the plot, but it’s funny and worth reading anyway. 
Here are some audio files that are quite nice, from one of my favorite composers.
I don’t know about fractal music, but noises that have power-law spectrums are fractal, in the sense that they’re scale-invariant. Example power-law noises are white noise (power is equal at all frequencies) and pink noise (power goes as inverse frequency).
While not music, many people find these types of noises soothing.
Quasi-related… this website allows you to turn Pi into music.
Wow, thanks for this link! WhyBaby loves these! She’s all wriggly and laughing, boogying down to fractal music - definitely a Doper in the making!