When using Safari at a place with a captive portal like Starbucks, iOS devices will launch a special modal webview to authenticate with the portal. I've seen third party apps trigger this so it's possible to do outside of Safari but I don't know how. It doesn't seem to come for free just by using Reachability or making a network request.
Edit: The best I can say is the answer I accepted works sometimes. It's probably the right answer but I don't know why it's not consistent. It can take several network attempts before it will launch the captive portal.