Okay, for those of you struggling with this website opening behavior instead of the game launching as expected.
After some painstaking searching on the topic, I found the following (forgive me, gog won't let me post links)
On that page, it directs you to look at your registry. If you don't find battle.net in either of those two locations please continue. If you do find it, well, I'm not sure where you should go from here.
Apparently my install of Battle.net on my D: drive isolated itself from windows. Indeed, not only could I not find it in the registry, but I also could not find Battle.net in the start menu even. I could launch it because I used the laucher.exe file inside the D: drive Battle.net folder, and pinned it to the taskbar and never thought anything was wrong.
So I tried a reinstall of Battle.net. Except when I did, it just "reinstalled" it over my current install and it basically did nothing. Just reopened the Battle.net launcher. So of course when I went back to GOG and hit "Launch" I got the dreaded website window.
Onto nuclear option then. Since my windows install literally did not recognize Battle.net as being even a program it ran outside of the icon on my taskbar, I went to my D: drive in explorer and deleted the entire Battle.net folder. Now, my blizzard games do reside in D: as well, the launcher is instructed to load them there. I did not have to delete those games, just the Battle.net folder.
Once deleted, I then ran the installer again, and this time installed Battle.net to C: drive. At this point I just wanted it to work, but I suspect that when the installer asked me where I wanted to install it, I could have pointed it to D: again and it would have put the correct registry entries into windows (ie, Windows would know that it does indeed exist).
Anywho, installed the launcher to C:, booted it up, insured that my games were still there (they were) and that under settings the launcher was still correctly pointing to the files where those games were located in the D: drive (it did). Then I shut the launcher down, pulled GOG back up, whispered a prayer ("Hold onto your butts") and hit the "Launch" button under Overwatch.
Worked like a charm. The launcher appeared, auto launched into Overwatch (Thank you, Sam Jackson).
TL;DR: If it directs you to the website to install, perhaps something is wrong with your installation. Check it, then maybe try and reinstall or 3 to see if it corrects your problem.