First of all, he can still burn DVDs without iDVD, there just isn't a standalone app for it anymore. Leo doesn't really think backing up to DVDs is the best way to go. They decay imperceptibly, so he would have to check them regularly to make sure they're still good. Leo says backing up to an external drive is much faster and better, and then also to the cloud using a service such as Carbonite. That way he has three copies and in two different locations.
Leo says it really depends on how the disc was burned. It's likely the disc has gone bad. There are two ways a disc can be burned. One is to burn the whole thing and finalize it, meaning it can't be altered later. The other is using UDF, meaning he would be able to add to it later. He would need a special driver to read that, and he'd have to finalize that disc before it could be read by other computers.
Since Walter doesn't have DVD menus and extra features to contend with, he can just rip the DVD to his hard drive and re-author a disc from that. There's a program called DVD Shrink which can put commercial DVDs on his hard drive so he could both create a new DVD of the content and play it from his hard drive. Leo doesn't think he would need this though.