@Allan Wright Have you tried repairing permissions as per my comment above?
I seem to have fixed this on my system by performing a permissions repair;
1. Open Disk Utility
2. Select main HDD partition (usually Macintosh HD)
3. Select the First Aid tab.
4. Click Repair Disk Permissions (or perform a verify first to see what's wrong)
Mine reported problems with /Library/ and I suspect this is what was causing it.
The problem seems to be specific to my user. If I login to OS X as a different user I can start Minecraft in Offline Mode. But what is it about my user that's causing the problem?
For me, this only happens on one of my Macs, my MacBook Pro. It's fine on my iMac.
I have got it to work once. I was force quitting and restarting it over and over again and one time it started. But then the next time it went back to the white screen.
I've tried reinstalling the launcher and removing the ~/Library/Application Support/minecraft/ directory, but it still happens.
What else do I need to remove if I want to do a completely clean install?
But if everyone else is experiencing what I'm experiencing, then it's an issue with the filesystem local to an OS X user account, nothing that Mojang can fix. I proved that I had a problem with permissions in my filesystem, there was nothing wrong with Minecraft. I must admit, it would be nice if the launcher kicked out an error message, e.g. "Failed to open file blah.blah," which would help you resolve this issue.
To anyone experiencing this issue I would suggest trying the following;
1. Login as a different OS X user and try running Minecraft. If it works, try repairing permissions via Disk Utility and try again with your original OS X user.
2. If it doesn't work under a different user, try re installing Minecraft (first backup ~/Library/Application Support/minecraft/)
3. If you get no joy with 1 or 2, try repairing permissions.
If nothing above fixes the issue, then there's a real chance that it's a bug in Minecraft.