Hi there! Dr. Hurst here.
I don't believe this is likely to cause a problem for your dog. The amount of zinc in the tube of medicine is minimal and gastrointestinal absorption is limited. Additionally the ingested medicine will pass through the intestines quickly. I have known a great many dogs and cats that have ingested Bacitracin with no ill effects. There may be some tummy upset, but nothing more. I would be more worried about the packaging she or he may have ingested.
Zinc toxicity is common with ingestion of pennies, and occasionally can be seen from ingestion of materials with a high concentration of zinc, such as zinc oxide ointments.
That being said, if you notice anything unusual, such as lethargy, increased drinking or urinating or vomiting, you might be well-advised to have your dog checked out by your veterinarian.
If you are worried and want something done just to be safe, you can take your dog to the emergency veterinarian and they can induce vomiting (if the ingestion was within the past few hours) an/or administer activated charcoal to absorb the medicine and prevent absorption by the intestines.
Best wishes and let me know if I can be of further assistance!
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