Hello, my name is***** and I have over 20 years of experience as a veterinarian. I’m sorry to hear that your fellow ate a chicken thigh bone.
While I understand that getting that bone out would be nice I do not recommend inducing vomiting in cases where dogs eat sharp objects like bones. The chances of tearing his esophagus or stomach as it contracts to force the bones out is much too high. I hope he at least chewed a little before he swallowed as that will make passage a little easier.
I understand that he is gagging a bit, likely because the bone was rough on his esophagus, but let's try to soothe it and eas ehis gagging.
In this case we need to try and hope that we can get this bone to pass through his gastrointestinal tract. And that his stomach acid helps us by dissolving the fragments.
To help I recommend starting with a vaseline sandwich. Take a half slice of whole wheat bread, slather with vaseline as you would peanut butter on a slice of bread, cover with the second half slice. Feed it to him bite sized piece by piece or he will make a huge vaseline mess. This adds fiber and hopefully makes the bones pieces slip rather than catch.
For the next few days, until the bone pieces pass, he'll need to be fed a soft, bland diet with lots of fiber added in small meals several times a day, 4 to 6 meals is ideal. We do this so the bone pieces are surrounded and the gut isn't irritated just contracting on bone.
A homemade diet for this is 1/3 boiled hamburger or white, skinless chicken, all fats and juices drained and 2/3 white rice. Add 1-2 tablespoons of canned pumpkin, not pie filling, just pumpkin, to each meal for fiber.
You'll need to check his stools frequently for bone pieces in the next few days.
Signs that things aren't going well, and he needs an immediate veterinary visit, are continued vomiting or gagging, a tense painful belly, lack of an appetite, a fever (more than 103.5F rectally) and lethargy.
If you aren't seeing bone fragments in the next day or two a veterinary visit and an abdominal radiograph to see where the bones are an option. If it doubt at all that things aren't going well it is always best to have him checked.
Once the pieces are passed and he's feeling well then start mixing in his regular food, adding a little more regular and less bland at each meal. It should take a week or so to get him converted back to regular food.
Let me know if you have any questions.