There are a lot better websites to ping for this, that are used for captive portal detection. E.g. Chromium uses http://clients3.google.com/generate_204 and Firefox uses http://detectportal.firefox.com/success.txt.
I mean I know this is google we're talking about, their uptime is second-to-none, but that still seems like a hell of a dependency.
But hey, give it a whirl: block detectportal.firefox.com with your hosts file and see what happens.