I installed the game via GOG Galaxy and when I select the GOG version of Diablo it opens a black window and crashes a second later.
Diablo (Classic) runs fine but it's streched to widescreen and I can't get it to run in 4:3.
Has anyone found a solution for this?

Running the game via DevilutionX should solve both issues for you:
https://github.com/diasurgical/devilutionX/releases

Note that if you prefer 4:3 over widescreen (not stretched) then you will need to change the diablo.ini settings.

Dont forget to recheck game files. Twice. But first make sure Galaxy always started "as admin" (and so - all games from launcher).
Than properly set GOG DX wrapper DXCFG tool. Play Hellfire DX version.

If your Windows 10 still crash game after set DXCFG (and you use latest stable updates build for Win10 and every solutions from support site for each Blizzard game - some very useful info there) - try rename ddraw.dll file. But that disable DX wrapper and so - that same as Classic laucnh option in launcher.

I got it fixed by simply restarting my PC.
Thanks anyways :)