If I've ruled out physical problems, perhaps consult with a veterinary behaviorist about anxiety or other issues not related to his physical health. I'm serious about this.

Is he crate trained? Perhaps he needs to just learn to "hold it" more? Keep him in the crate a couple hours a day to help him learn this.

Finally, if nothing else works, get him a belly band. Very cheap and effective for male dogs. Put in a pad, velcro it on him. We had a dog that did something similar. He had to wear the band when we were not home. 90% of the time he didn't go in it. But sometimes he would and it would catch the urine. Toss in the wash, as needed. Done.

I would (1) talk to the vet and (2) get him a belly band or other diaper belt.

Is he neutered? I recall hearing that intact dogs do this more.

My one year old dog leaves tiny droplets of pee in the house, the number of which varies per day. They're hard to see on the floor and it's impossible to catch him in the act (he just walks about without squatting or lifting).
He tested negative for a UTI.

This is what I've noticed, since I work from home: he tends to leave drops more often -
1. in the entryway, when coming to investigate someone coming in or out, or expecting a walk;
2. when he's left alone in the house;
3. when he gets excited because the kids are excited.
But sometimes I'll come across a drop and none of these 3 events above will have happened that day!

It's frustrating! He didn't use to do this, but can't seem to correct himself, either because he doesn't feel it himself, or because I discover the pee too late for him to make a connection, even if I express displeasure. I don't know!

What would you do?