How do you remove the rear view mirror on 2015 Honda Civic?

It keeps dropping down so I need to replace it.

Look for a 2-3mm hex or Allen grub screw on the base that connects to the window anchor.
Loosen it much further than seems necessary so you won't pull the anchor off the window. It should slide off then.

I think it slides down on it. At any rate, examine the interacting parts and see what you can figure out. I don't think it is broken.
The one real answer for you is to visit a junkyard where you can get a replacement. Look at a manual to find out how it is keyed into the frame.
I took off mine on a VW and it is twisted to take off, but I think honda slides down on slides.
Of course, if your metal plate glued to the windshield comes lose (which it doesn't sound like) then you need to get a kit to clean the glass and glue the chip back up. You have to do a good job.

Using an angle grinder will remove most things