Hello from JustAnswer,
There are a number of possible causes for a week of diarrhea without fever or vomiting, but the two more likely considerations are infection or food allergy.
Regarding infection, although an intestinal virus is certainly possible (e.g., norovirus), as a pediatrician, I would strongly consider evaluation for a bacterial infection if the diarrhea lasts more than 7 days (e.g., stool bacterial culture and C. difficile bacterial toxin assay in addition to rotavirus test, ova and parasite test, and Cryptosporidium stain test). If no infectious cause is found, then evaluation for a food allergy may be considered. If that also yields nothing, then referral for evaluation by a pediatric gastroenterologist (digestive system specialist) may prove helpful.
Thank you for the extra information.
I have treated a few young patients who had no fever, vomiting, or abdominal discomfort but were finally diagnosed as having prolonged loose to watery bowel movements due to intestinal infection with Clostridium difficile bacteria (as mentioned in the series of tests in my prior post). I mentioned norovirus in my prior post because I have seen patients who were smiling, playing, and generally appeared well except for the diarrhea due to this intestinal virus. If he has recently started taking cow milk or soy milk, then sensitivity to the protein in either of these liquids may be the cause of decreased appetite and diarrhea. If infection, food allergy, and every other serious condition is ruled out, then a diagnosis of toddler's diarrhea may be made (discussed in the reference below).
Malabsorption reference: http://emedicine.medscape.com/article/931041-overview
With some cases of diarrhea due to viral infection, I have seen prompt clinical improvement with a daily dose of over-the-counter "Culturelle for Kids" probiotic mixed into cool to room temperature liquid or food and given until the diarrhea fades. Again, if your son's diarrhea lasts beyond 7 days, please seek medical evaluation for a possible bacterial infection.