Friend, certainly vomiting can be caused by viruses or bacteria. But, if the other cat did not exhibit these symptoms unlikely for this to be something contagious. Other possible causes are: Ingestion of a foreign body, toxicity, pancreatitis or even part of a systemic disease (kidney or liver disease), etc. to name a few causes.
I would recommend a hands on examination by family vet; he may need some testing (blood work and xrays) to determine the exact underlying cause. Treatment will be based on exam and test results. In addition, he may need supportive care (IV fluids, anti emetics, etc.). In the meantime this is what you can do:
1- Do not give things like Pepto Bismol or imodium. Start him on Pepcid AC (famotidine) 10mg 1/4 tablet every 12 hours for the next 2-3 days. You can buy Pepcid AC over the counter in any pharmacy (chemist); if you can't find pepcid ac just leave him without anything.
2- No food or water for 6-8 hours. Need to give his gastrointestinal tract time to rest.
3- After 6-8 hours offer a small amount of water. Wait half hour, if no vomiting, offer small amount of rice (50%) and boiled boneless chicken or ground beef (50%) 4-6 times a day. Keep on this diet for several days then wean into regular diet.
4- Follow up with your family veterinarian for a hands on examination.
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