If your pet has a medical emergency, please contact a veterinarian immediately.
The solution is to express the bladder to ensure he is empty. This video is really good because it shows several situations for expressing a dog's bladder. One of the situations is a senior dog who simply does not empty completely. They demonstrate how they let the dog begin urinating, then they apply pressure to help him finish, so he gets empty. Teamwork. It is demonstrated standing, but you can do it with him lying down, too.
Please don't feel too stressed about this. Urinary infections normally clear up pretty quickly with common antibiotics. He looked like he was emptying normally and the vet did not check whether he was, or tell you to express, so the vet should not be negative to you about this.
In addition to the Scout's House video above, this article has information about expressing and a long list of videos at the end according to the size of the dog.
Several people here have recommended Solid Gold Berry Balance, but I haven't tried it.