it depends on the type of background check that is being done...

a general criminal history background check... no not unless you were arrested or transported to the hospital by the police, sheriff department for involuntary assessment and commitment. it will only list transport to hospitals name for the hospitals side of the record. it will not contain direct info about what happened at the hospital.

a general credit history background check.... yes if you, (not your insurance ) paid or didnt pay your hospital bill, not paying hospital bills results in the bill entering collections and placed on your credit report as an unpaid bill which decreases a persons credit rating.

medical background check.... yes it will show up that you were hospitalized in what ever hospitals name is (thats not protected by privacy laws, what is protected by privacy laws is why you were there. background checks can not get into your hospital records to find out why you were there)

mental background check... yes it will show up that you were hospitalized in what ever hospitals name is. it will not show what went on at the hospital.

some locations just do the general criminal background check (for apartment applications some job careers...) and others do the full background check.. (applications to a city or government job, adult/ child protective services on abuse cases, ....and others)

my suggestion is find out which type of background check someone is doing for you.. they will tell you whether they will be contacting area hospitals for verification of whether you were admitted (which is part of public records ) and why they need that info.