Fortunately, IE7 did provide a nice diagnostic tool that decided that the ports for FTP, HTTP and HTTPS were all unavailable and suggested that I investigate my firewall settings. Interestingly enough, it turns out that I'd disabled Windows Firewall some time ago, because I trust my dedicated router/firewall to keep the black-hats out of my home. Turning to my work laptop (which could still access the outside world), I quickly found a web page that strongly recommended leaving the firewall on during the install. Need I say that I wasn't very surprised by the implication that IE7 can break other programs' access to the Internet?
The good news is that uninstalling IE7 fixed everything. Until I can get more information about what happened, I've told Windows Update to not try re-installing IE7.