From a physician’s point of view, I can tell you that pets do generally improve your health, especially dogs. To me, one of the greatest benefits they bring is keeping you fit and active. In a study of dog owners published this year by BioMed Central, dog owners typically walked an extra 20 minutes each day, the recommended length of a normal workout.
Thus, if the president had a dog and didn’t delegate dog-walking duty to an aide, he would be walking outside more. The exercise would help to keep his weight in check. At the same time, he would definitely improve his heart health by lowering his blood pressure and even controlling some of the blood chemistries.