Hello & welcome, I am Dr. B, a licensed veterinarian and I would like to help you with your wee one today.
Since this has not settled in the past week, we do want to start taking some steps to help him at this stage. First, you can try a light/easily digestible diet. Examples of this would be boiled chicken with rice, boiled white fish with rice, boiled lean mince with rice, scrambled eggs (made with water and not milk), or cottage cheese with rice (you can substitute rice for another starch like plain pasta or potato but we find rice is quite well tolerated by dogs. There are also veterinary prescription diets that can be used in cases of gastroenteritis (ie Hill’s I/D or Royal Canin’s sensitivity). The easily digestible diet will be better tolerated and absorbed by the compromised gut and should get some nutrients in and result in less diarrhea. Also feed this as small frequent meals to further decrease the volume of diarrhea he is producing. As long as improvement is being seen, I usually advise that the diet be continued until the diarrhea is settled, and that they are then slowly weaned back to their normal diet over a week. If he isn't up to date on worming, it would be prudent to cover your bases and at least rule out the GI worms that could be initiating GI irritation. To treat worms, you can buy worming products over the counter at your local vet or pet stores. There are a range on the market, but you want to use a good quality wormer that covers both round worms and tapeworms. In this situation, it would be ideal for you to treat with Panacur, Drontal (LINK) or Milbemax (LINK) as it will cover all the worms in question. Do make sure to have an idea of his weight before purchase wormer to make sure you get the correct dose for his size. Now since it has been a week, I do want to note that diarrhea can dehydrate a dog. Therefore, if want to trial home treatment with him then do keep an eye on his drinking and hydration. To check his hydration status to make sure they are not becoming dehydrated there are a few things we can test. One is whether the eyes appear sunken, if the gums are tacky instead of wet/moist, and whether he has a "skin tent" when you lift the skin. To see how to check these parameters for dehydration, you can find a wee video on this HERE. If you are seeing any signs of dehydration already, then you do want to have him seen by the vet before makes him feel poorly. (since it is often the dehydration that starts to tap their energy level and actually makes them feel ill)
If you were concerned that he was becoming dehydrated, you can try and encourage him to drink but offering fresh water or even low-sodium chicken broth. If he isn’t amenable to these, you can syringe feed pedialyte. Pedialyte is nice (though aim for a flavorless one) because it will get some of those lost electrolytes back into him as well. A typical maintenance rate for hydration in an animal is 48mls per kilogram of weight a day. If you do give syringe pedialyte, this should obviously be divided up into multiple offerings through the day rather then all at once. This value will give you the total he needs for the day and is a good starting point to give you an idea of a dog's daily requirement. (we aren’t calculating losses, so you can add an equivalent volume to match how much diarrhea is being producing). If he vomit when you have given pedialyte, then therapy should be discontinued (since we don’t want vomiting because of our intervention).
To slow the loose feces and give him a bit more 'control,' you can safely give him Protexin Pro-Fiber (example) which you can buy over the counter from your vets. It will slow loose stool and firm it up, but also has pre & probiotics to restore GI balance. Otherwise, Kaolin/Kaopectate (LINK) which is available over the counter at most pharmacies.
Overall, there can be many triggers for diarrhea of this nature. Since he is young (and doesn't have the reserves of an adult) we do need to be proactive in trying to settle this form him. Therefore, do try the above supportive measures for him. If he has just had a wee dietary indiscretion or if GI worms are to blame, these steps should settle this for him. But if you do so but his signs do not settle, then it would be worth following up with his vet. You can even consider having them submit a fresh fecal sample to the lab to be checked for common parasitic, protozoal and bacterial causes of diarrhea. And if this is a bacterial or protozoal complaint, knowing what the causative agent is will help you treat it effectively and clear this abnormal stool to get him back to passing normal feces.
I hope this information is helpful.
If you need any additional information, do not hesitate to ask!
All the best & happy holidays,