So I've downloaded the Witcher 3 four times now, and I'm clearly missing something obvious. I've installed the game with GOG Galaxy this time, the play button is green, but I can't play. I click the play button and nothing happens. I do not know what I'm missing, so I'm humbly asking you all to point out my stupidity. What steps am I missing?

I have the exact same problem. Launching the game from galaxy doesn't work, but using the actual witcher3.exe works. It used to work but i've been having problems getting it to launch from galaxy for a few days now.

After two days I figured out what was going on with mine. I had to go in and repair the c++ package. You can find that under your Witcher 3 folder _redist/MSVC2012_x64. That's where it was for me, anyway. Repairing that finally let me play the game. Not sure if I'll have to continue to do that in the future or not. Either way, super weird bug and I'm glad it's fixed. Hopefully that will help you as well.

Have you contacted support?
Have a look here:
http://www.gog.com/forum/the_witcher_3_wild_hunt/faq

Make sure you've downloaded the patches as well. I don't know for certain, but not having 1.0.2 installed may cause that issue.

download and install this https://www.gog.com/downlink/the_witcher_3_wild_hunt/en1patch1


then this one https://www.gog.com/downlink/the_witcher_3_wild_hunt/en1patch2

and finally https://www.gog.com/downlink/the_witcher_3_wild_hunt/en1patch3