Based on the way the system works as I am writing this, the answer is no. The contents of the profile page are the same on every site. (Well, technically, there are two profile types, one of which is getting phased out and replaced by the other, but neither one has custom fields.)
It's just that nobody's ever built any customization options into the design. I can't really imagine us making a change just for CiviCRM; it seems like too specific of a use case to justify the work involved. This might be something that we could look at if other sites also wanted custom fields for some reason, but I'm not aware of any off the top of my head.