I'm using a Macbook Pro with OS X version 10.9.5.

When I click the icon, a window pops up immediately, saying that Krita quit unexpectedly. I click Reopen and the window just pops back up. It doesn't even TRY to start up the program. My security and privacy settings aren't interfering with it; the program just doesn't open AT ALL.

That's a very unspecific problem description.
Krita gets heavily tested, before it's released and is therefore assured to work.
If it crashes for you there are a ton of possible reasons outside of Krita itself.
After crashing there should be a problem description somewhere. Sadly I'm not too familiar with MacOS.

10.11 is the oldest OSX version that Qt still supports.: see http://doc.qt.io/qt-5/supported-platforms.html .

Open the terminal app and try to start krita from there; maybe that will give some more information. If you've copied the Krita app bundle to /Applications, the command is like this:

/Applications/krita.app/Contents/MacOS/krita

But it's likely that we cannot support 10.9 (or 10.10) anymore.