No. I recently contacted a SE representative with the same question, and they said absolutely not. They did recommend I make a suggestion to SE about it though, so maybe someday in the future we can, should they allow it.
I'm not disputing the above answer, because I'm not familiar with "profiles" as related to FFXI. I do know you can create a new handle and transfer or re-link your content ID's to that, that way anyone in your F-list won't see you online anymore without doing a /search, because they're F-listed to your old handle.