Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that your pup ingested a bone on Thursday and vomited up a chunk of it today.
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 understand that you didn't induce vomiting, but he brought the bone chunk up without your intervention. The concern though is that he may have done some damage in bringing the bone chunk back up.
How is he behaving now? Was there any sign of blood in the vomitus or is he gagging and uncomfortable now?
If he seems very comfortable now and there was no sign of blood in the vomit, and he is eating and drinking normally he may not need an emergency visit, at least not now.
In this case we need to try and hope that we can get the rest of the bone to pass through his gastrointestinal tract. And that his stomach acid helps us by dissolving the remaining 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 bone pieces slip rather than catch.
For the next few days, until the remaining bone pieces pass, he'll need to be fed a 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 contracting on just bone.
A homemade diet for this is 1/3 boiled, lean hamburger or boiled, white, skinless chicken, all fats and juices drained off the meat, and 2/3 boiled, plain, 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, a tense painful belly, lack of an appetite, a fever (more than 103.5F rectally), bloody diarrhea or black tarry stools, 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 bone pieces is 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 to get him converted back to regular food.
Let me know if you have any questions.