Could a broken tire belt cause my care tire to vibrate?

Absolutely, broken belts throw off the balance of the wheel. Look for buldges or indentations in the tire. If the vibration is felt in the steering wheel, it is probably a front tire. If the vibration is felt in the seat, it is probably in a rear tire. Most manufacturers will pro-rate a replacement tire for the manufactured defect. Cut tires, curbed tires, under-over inflation wear could also effect the warranty. Vibrations could also be caused by other mechanical defects: lost wheel weight, bad wheel bearing, bad ball jounts, bad shocks, sticking brakes, etc. Rotate the front tires to the rear, if the vibration changes in the steering wheel, then chances are the problem is in the wheel/tire.