Gallbladder inflammation is also known as cholecystitis. Antibiotics are an unlikely cause of gallbladder inflammation. There are several points that need to be clarified here.
The feeling of being bloated and burping does not mean you have cholecysititis. The diagnosis of cholecystitis is usually made by history, physical, labwork and imaging. The most sensitive imaging test for cholecystitis is an ultrasound
of the right upper quadrant of the abdomen, which looks for presence of stones inside the gallbladder, thickness of the gallbladder wall and fluid around the gallbladder. These are all common findings in acute cholecystitis. There are other causes that can cause the symptoms you describe. The most common symptoms of cholecystitis are right upper quadrant adbominal pain, nausea +/- vomiting
and fevers. Burping and bloating are generally non-specific.
As far as your question about whether or not antibiotics can cause cholecystitis, it is important to know which antibiotic you took. There are certain cases where Ceftriaxone has been reported to cause gallbladder sludge formation leading to cholecystitis (inflammation of the gallbladder). This is not very common and even though it is reported in the literature, it should be very low on the index of suspicion as the cause of cholecystitis -- that is IF you truly have cholecystitis.
The more likely cause of your burping and bloating is the fact that many antibiotics can irritate your stomach causing the symptoms you describe.
Ultimately, your physician has the best idea of how to proceed with your workup, because they are the ones that can perform a full history and physical exam which would be key in diagnosing a problem.