Our population grows at an exponential rate. Strictly speaking, we're not overpopulated now, but in a few generations we will be. We probably won't run out of room before food production can no longer keep up. We may find a more efficient way to produce food that can keep up, but the fact is that this will only slow down the problem. When we do run out of room (or approach that point, anyway), illness will spread quite readily, the end result being plagues, regardless of our collective hygiene. We could solve this by having regularly scheduled mass death events every few hundred years, such as nuclear wars, or maybe we can get playful with it and do some kind of gladiatorial games. Perhaps better yet, is to just get a large portion of our population to colonize new worlds. That's perhaps the only practical solution that doesn't directly involve people dieing.
This solution doesn't address the environment. This is not meant to be a solution for the environment. This is a solution for human kind. There's no reason why we can't continue to develop environmentally friendly technologies while working toward colonizing other worlds. In fact, I'd say the goals are quite compatible, as technologies which require fewer resources will contribute nicely to founding new civilizations.