There were some British studies a few years back that showed larger admissions to emergency rooms and such, but came to no conclusions. There is a minor statistical correlation, but no really hard data that would explain it.
The thing is, the moon isn’t particularly closer during a full moon, it’s merely a phase of light reflection. The changes in gravitation attraction that cause the tides are daily cycles, not monthly cycles. If it was simply a light thing, you’d expect the same type of rise in hospital visits to occur as the days get longer in spring and summer, too, but there is not stats to back that up.
It’s kind of odd, probably just psychological.