HAV is mostly diagnosed depending on detection of antibodies specific to the virus in the serum. it is transmitted by fecal-oral transmission and the virus excreted in the stool, routinely it is not diagnosed by antigen detection in stool.
The serology test for HAV is quite good, and ELISA on stool has some inherent problems as feces contains lots of componds and enzymes that may influence the analysis. However, both calprotectin and H pylori can be detected in stool samples by ELISA, so it should be quite possible to do.
Not practical as per the 1st answer. The best way to identify HAV in stool is by PCR. The advantage is that HAV excretion in feces is way longer than viremia, therefore that is a useful substrate when HAV PCR in serum is negative especially for outbreak investigation.