Ezekiel, I'm not sure if this is an issue for 1.6.2, but I tried using the demo to make sure that Minecraft will play when I purchase the game in earnest. So, I tried using Safari and Firefox (Chrome is crap for not enabling Java) for the browsers on my MacBook. And they won't work with the demo. The demo runs on an older version of Minecraft, v.1.5.2. If and whenever I do find a bug if I ever purchase and download the latest version, I'll make sure to alert you all and Mojang here!
Anon Ymus, I updated Java to the latest version. The Mac App Store updated Java there, and I went ahead and downloaded the latest version of the launcher from Java's website, but all I end up with is the crash alert when I try using the demo to make sure Minecraft will work fine on my MacBook Pro with Retina Display (15-inch, Ivy-Bridge Intel Core i7 2.6 GHz, 512 GB SSD, 16 DDR3 RAM, NVIDIA GeForce GT 650M with 1 GB VRAM). I switched over to the PC partition of my MacBook, running Windows 7 Home Premium, and after updating the Java launcher, the demo works fine, but I use Win7 for storing documents and working on Microsoft Office apps. And I only have 1.2 GB of storage left after partitioning my SSD with 50 GB dedicated to the PC partition. So I want a copy on the Mac so I can have enough space to run the game and have saved maps. I was only using the demo (v.1.5.2) not the latest version (1.6.2). If the Mac is able to run 1.6.2, please let me know!