At the risk of stepping into the middle of a fight… I must suggest that anyone interested in the relationship between fatness, fitness, and health read a fantastic, complete, and highly accessible scientific review on this subject written by Vojtech Hainer, Hermann Toplak, MD, PHD, and Vladimír Stich, MD, PHD. For the TL;DR crowd:
1. It is inarguable that being fat and sedentary is terrible for your health.
2. Quite a few recent, large, and well-crafted studies have dug into figuring out which is more important for preserving health: being normal weight or being active. Almost all—and particularly the better crafted studies that have used independent and scientifically rigorous measures of fitness rather than surveys—show being active is the more important factor, even if the activity does not succeed in causing one to lose weight.

The numbers in this figure represent the relative risk of dying in a given year, broken down by fitness and fatness; lower numbers are better. The healthiest group in this study were the (moderately) fat, fit group.
A prospective JAMA study from 1995 showed nicely that becoming fit had a significant effect on both all-cause mortality and cardiovascular mortality. The effect size wasn't small. For every minute one could last longer on a treadmill, the relative risk of death in a given year was reduced by 7.9 percent.
3. Therefore, just because someone is fat (or looks fat) doesn't mean they're unhealthy. Additionally, even if exercise didn't work for you (i.e., it didn't make you thin), you are still deriving a massive benefit from exercising even if you remain fat. The amount of exercise needed is quite small, 20 minutes of moderate exercise (enough to get your breathing rate up, not to drive you into a drenching sweat and gasps for air), three times a week is enough.
Updated:
Here is data for women, around the same question. This study isn't quite as clean as the one for men I cited above.

In this study—the Lipid Research Clinics Study—both fatness and fitness were relevant predictors for health. A more ethnically diverse population of subjects was involved in this study. The results from slightly overweight and massively overweight individuals were combined, perhaps clouding the relationship in the other (men-only) study was able to demonstrate.
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Every Size volunteers had kept their weight stable, neither gaining nor losing a significant number of pounds. In contrast, the dieters had lost weight by the sixth month, but regained it by the 2-year checkpoint. Their beginning weights and their weights 2 years later weren’t significantly different.
At the start and end of the study, total cholesterol and systolic blood pressure were in the normal range for all the women. Within this range, however, the Every Size women lowered their total cholesterol and their systolic blood pressure and were able to maintain those reductions for the entire course of the study.
In contrast, the dieters didn’t lower their total cholesterol at any point in the study. And they weren’t able to maintain the healthful decrease in systolic blood pressure that they’d achieved just after the 6-month reducing-diet phase
What about physical activity?
At the 2-year point, Every Size team members had nearly quadrupled the amount of time they spent in moderate, hard, or very hard physical activity, compared to what they had reported at the study’s outset.
The dieters didn’t fare as well. At the 1-year point, they were exercising more than at the start, but they didn’t sustain their improved level to the 2-year checkpoint.
Although all the dieters made a lasting improvement in at least one of the food-related habits called “eating behaviors,” the Every Size volunteers improved in more of the categories.
The researchers also monitored depression, a common problem among large-sized women whose low self-esteem may be related to their body image. Both groups made significant strides in lessening depression, but only the Every Size women were able to preserve a more optimistic outlook.
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However, the recent meta-analysis of studies on the relationship between physical activity and abdominal fat using imaging techniques revealed that reductions in visceral and total abdominal fat may occur in the absence of changes in BMI and waist circumference (41).
Thus, insensitivity of anthropometric indexes to reflect actual amounts of visceral fat may contribute to overestimating the role of fitness in relation to fatness in their effect on cardiometabolic risks.
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Federal guidelines call for 150 minutes of moderate to vigorous physical activity every week, including two days of full-body strengthening.
(snip)
...it may be time to consider more drastic measures, from employer-mandated exercise breaks for office workers to auto-free zones that encourage more walking.
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