OK, now this is getting rather specific, eh?
It's not National Ice Cream Day (though it is National Ice Cream Month), it's National Peach Ice Cream Day. So, if you don't like peaches than you're out of luck and you have to eat broccoli all day long.
This is timing that actually makes sense. Sometimes you don't know why a certain holiday falls on a certain day or in a certain season, but it's summer so having a day set aside for an ice cream flavor seems right, and peaches are a summer fruit and sound awfully refreshing.
P.S. This past Sunday was National Ice Cream Day??? How the heck did that go by unnoticed?