Yes and no, though perhaps most importantly no. Saying that something is an art is sometimes a way of saying that it's an exercise of skill, not least of a skill that isn't simply a matter of following a set of instructions. In that sense, cosmetic surgery is an art. Cosmetic surgery also has an obvious aesthetic dimension and no doubt calls on many of the same skills that a good sculptor needs. So all of that is on the "yes" side.
But there's another obvious sense in which cosmetic surgery isn't an art, or better, perhaps, an Art. Painting, sculpture, poetry, etc. are Arts in this sense not just by virtue of being skills whose practitioners may have aesthetic goals. They also fit into a familiar set of cultural practices and institutions (museums, galleries, performances, reviews, critical studies, sales, auctions...) that determine what we count as "Art" with a capital "A." Cosmetic surgery isn't an "Art" in that sense, and this is almost certainly a very good thing.