Food can be pervasive in Jewish culture, as it is in many cultures. However, gluttony and sloth are not honored in Jewish tradition. In fact, the opposite is the case.
In addressing your question, I can’t possibly improve upon the following statement of Rabbi Moshe ben Maimon (Rambam; Maimonides) in his great code of Jewish law, the Mishneh Torah (HilkhotDe`ot 4:1). There, he teaches that keeping our bodies healthy is an integral part of our worship of God, “for it is impossible for one to gain any knowledge of the Creator when one is ill.” Therefore, he continues, one must abstain from substances and habits that weaken the body and behave in ways that are conducive to physical health. He follows this with a detailed prescription (remember that Rambam was a physician) for a proper diet, good sleeping habits, sufficient exercise and prudent personal hygiene. Given that Rambam was working on the basis of the best available science in his time – the 12th century – it is quite possible that we today would disagree with some or many of his instructions. His basic point, however, is as relevant as it ever was: the preservation of bodily health and well-being is an essential element of Jewish religious teaching. So, to respond directly to your question: yes, there is a Jewish obligation to look after one’s diet, exercise, etc. Or, to put this differently: there is no mitzvah that requires us to overeat or to eat an unhealthy diet.