Who told you to give baby aspirin? Puppies don't have the enzymes necessary to process the aspirin.
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Aspirin is poorly tolerated by young dogs, since they lack the enzymes necessary to process the aspirin. The same is true for most cats.
Aspirin can cause gastrointestinal upsets and ulcers in dogs, just as in humans. You should pay attention to your dog's eating habits when you have been giving it aspirin to watch for signs of an upset stomach. If there are any signs of ulcers, such as blood-tinged vomiting, it is important to stop the aspirin. Since aspirin slows down blood clotting, a bleeding ulcer could prove fatal to the dog. Giving Your Dog Aspirin - Succeed with Healthy Animals: School for Champions |
While teething can causes lots of problems in human babies, dogs don't really have the same degree of trouble, and giving aspirin is an unnecessary health risk, and it's probably causing the vomiting. Try pounding the food with a hammer to crush it, or moisten it with water.