Yeah, generally speaking unless it's a very large shard and cuts your esophagus, once it reaches your stomach, acids will knock the edges off and it'll pass without issue.
Gastric acid is basically hydrochloric acid, and rather less concentrated than the standard laboratory bench HCl, which stands peaceably for years in glass flasks. I would be wary of assuming that chemical attack will quickly blunt glass shards.
I really doubt stomach acid would do anything to glass. Strong acids are regularly stored in glass.
HCl will do nothing to glass. It's strong bases that will dissolve glass (a little), and your digestive system doesn't have any.