I am having the exact same issue. I have a Motorola Droid 1 on Verizon. I noticed the issue on Thursday, August 25th which is persisting until today. It worked while I was in a mall and then it stopped working later on during the day. Google and google related sites will load, but other websites will not load. In addition, some apps will not updates or load properly (e.g. weather.com application). I have not tried using WiFi, but it does seem to be a localized issue with the 3G. I am not sure what is causing it, but I wanted to mentioned that the OP is not the only one. My assessment was that it must be something wrong with Verizon, as nothing has changed with my phone since I used it in the mall, but perhaps I missed something or a setting changed and I didn't realize.
I have tried turning on and off the phone, turning on and off airline mode, turning on and off 3G, clearing browser cache, using different browsers, but nothing has provided results.