If you consume fewer calories than you burn you will lose weight whether you exercise or not. Exercise makes things a lot easier though and helps ensure that your weight loss will be maintained. Not to mention that people who lose weight without exercise often find that they still aren't happy with the appearance of their body even once they reach their goal weight.
Weight loss, yes. Health? A bit of a different kettle of fish.
Weight (or weight loss) doesn't equate "healthy" body; you can be overweight and very active and have a healthier cardiovascular system, etc. Hence, the connection between diet modification and exercise is not just about accelerating calorie burn, but achieving (and maintaining) overall health.
If you are asking the question because you're just starting out and you can't face the thought of exercise yet....well, I think for a lot of people making the changes to diet/food is a very big first step and you could work on those habits first before starting to make activity changes too.
I agree with the others. Yes, you can lose weight without exercise. However, the amount of calories you consume will be a lot less. The more active you are, the more you can eat. If I were a little more active than I am right now (I walk, but I don't do strenuous exercise) I could probably eat about 100-200 calories more per day. Right now I can only eat about 1200-1300.
I also agree that you might want to start out with food changes first and keeping an eye on your calories. Then, as you lose weight, you might be more inclined to exercise.
One of the things that really helped push me along when I first started was doing Pilates. I purchased the "Pilates for Dummies" DVD and it's not anything strenuous, it's more for toning. However, even with the Pilates and the walking I sometimes do, I've never been 'religious' about exercise and have lost 80 pounds. But, it's a given, exercise is so good for you. Not only good for your heart and even blood pressure and cholesterol levels (among lots of other things), but it keeps you toned.
I agree that exercise definitely helps, especially in maintaining weight loss. But I read something interesting concerning which of the two, calorie reduction or exercise, is more important.
Overall, I think calorie reduction is key. What I read put it this way: it can take a couple hours of exercise to burn off a candy bar, whereas simply omitting eating it requires no further action on your part! Just not eating those extra calories is much simpler than trying to burn them off later (although having the will power not to eat the candy bar is easier said than done!)
It can be done, yes, but you miss out on a lot of advantages. Increased muscle mass from exercise (weight training) means your body burns more calories at rest, which helps (obviously) with fat loss. Exercise is also a mood-booster and supresses your appetite. Cardiovascular health is very important for long-term good health and longevity, and the only way you're going to get a strong heart and lungs is through exercise. Also, exercise defines and tones your body so you're not just "thin," but you're FIT. I know I've said this in another post, but I'll say it again here: you can diet and diet and diet and end up "thin" like Olive Oyl, or you can diet and exercise and end up with a fit, fabulous body. I guess it depends what you'd like to look like when you get down to goal weight.
First Goal: 200 lbs by Namibia trip (May 26, 2013).