I found what seems to be a tear in the driver side CV boot. Grease has leaked out near the middle and is just starting to spatter on the inside wheel rim and other surrounding components.

Is this a bad problem, something I should fix really soon, or just keep an eye on?

Well if you run it out of grease the joint will start to wear out and then you need an axle. If you do it right away you can get by with just an axle boot kit.