I understand that you are concerned that your fellow may have ingested spoiled meat.
The concerns here would be:
1) eating spoiled meat which can have high bacterial counts, like salmonella.
2) the other concern is that some hamburger is very high in fat and rich, and if he weren't used to eating meats with high fat content that could trigger a bout of pancreatitis and stomach upset.
Because the other two dogs are fine he was either unlucky and ate a pocket of meat that was spoiled or he is more sensitive to high fat foods. The other possibility is that the meat has nothing to do with his lack of appetite and stomach upset/vomiting.
Vomiting can be triggered by eating something they should not, too much table food, too many treats or something they find outdoors.
More serious causes of vomiting include viral or bacterial infections, chronic pancreatitis, a dietary allergy or sensitivity, inflammatory bowel disease, internal organ failure (kidney or liver disease), a full or partial gastrointestinal obstruction or even infiltrative cancers such as lymphoma.
To try and settle his stomach now you can give either:
1) Pepcid-ac (famotidine) at dose of one half of a 10mg tablet per 10 to 20 pounds of body weight every 12 hours.
2) Prilosec (omeprazole) at a dose of one quarter of a 20mg tablet per 10 to 20 pounds of body weight every 24 hours.
These will reduce stomach acid and could help settle his stomach. These can be used for several days as necessary as they are quite safe.
I would with-hold food for 12 hours to let his gastrointestinal system rest.
Small amounts of water, pedialyte or ice cubes to lick are fine to keep him hydrated.
Then if his nausea improves and he stops vomiting feed a homemade bland diet of 1/3 boiled, very lean hamburger or boiled, white, skinless chicken, all fats drained off the meat after cooking, and 2/3 boiled, plain white rice. Feed small meals several times a day.
Once he's feeling better 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 his regular diet.
If he continues vomiting, even after the acid reducing medication is given, refuses to eat or has explosive diarrhea it is best he see a veterinarian for an examination, injectable medications and oral medication like sucralfate to coat his gastrointestinal system.
Let me know if you have any questions.