Playstation Network is currently offline. The FFXIV launcher needs you to be connected to the Playstation Network in order to work properly, hence the black screen. Sadly you can't do anything to fix this except wait (I'm in the same predicament now.. reinstalling FFXIV on my laptop just so I can play on my day off!) It's been rumored that a group of hackers have taken down PSN for fun, although this can't really be confirmed until Sony talks about it