I can once again connect to Google websites with both of my two networked computers, after a lapse of a couple weeks. Apparently, the problem was in the hardware - specifically in the Linksys router. How that could happen (how I was able to connect to all other websites normally, but not to Google) remains a mystery.
My daughter succeeded in contacting Google about the problem, but its automated responses didn't provide any advice that I hadn't already tried. Their suggestions to view Google websites for further information are hardly of use to a person who's lost the ability to connect to Google.
Before I received the forwarded e-mails from my daughter yesterday, however, the problem went away - seemingly through divine intervention. Here's what happened: We had an freak neighborhood power outage yesterday that lasted two hours. I took advantage of the situation by turning off and disconnecting all of my DSL-modem and router wires from the phone line, computer, and back-up power-supply (UPS) device. When electricity to the house returned, I re-connected those wires and turned that internet equipment back on. After waiting and powering up the computer itself, I found, that I was once again able to connect to Google with my three internet browsers.
So, this appears to have been a problem resolved through hardware cycling. I had previously done this and even tried exchanging the DSL-modem with an available older one, but had apparently not done the thorough job that was necessary. Broni was on track with his suggestion to by-pass the router, but that attempted fix, for whatever reason, didn't resolve the problem at the time.