Merry Christmas is a standard greeting used over and over. Happy Fourth of July is not like that. You might say it to someone but it's not traditional (in a set way) like Merry Christmas or Happy New Year.
I agree that we don't have a fixed greeting. When I mention the holiday, I usually say "Have a nice 4th of July," referring to the celebration. You would usually say this to your co-workers when you leave work, and to other people you meet. (The celebration traditionally consists of a barbecue with family and friends, and public fireworks in the evening.)
The Vietnamese who operate my local doughnut shop were wishing each of their customers a "Happy 4th of July!" It sounded odd, but friendly.