Symantec is a very potential source for problems even if it's working, and even if you have thus far managed to avoid any problems caused by it. It is also a resource hog and slows up the Windows startup process to variable extent.
If I were you, I would download some free anti-virus software (like Avast! or AVG) that are less resource-intensive and AFAIK actually equally or more effective against infections than Symantec (and if you don't want to trust Windows firewall, get also some free firewall like Comodo or equivalent), then remove the ethernet cable for a while, remove everything related to Symantec from your PC, restart as prompted (multiple times), then install new AV (and firewall) software, hook up to Internets and see if your Windows security center wants to keep nagging.
Mind you, I have had some bad experience with Symantec - apparently the firewall component was jamming itself down so that it blocked almost all traffic - it made all browsers non-functional, but interestingly there were some things that worked perfectly, like torrents and SSH. And of course I couldn't just terminate the process because it was protected and crap. Only effective way to fix the problem once it started was to restart the PC, which got really annoying real soon.
Of course, I don't have Windows security center activated either, I removed it from the services activated at the startup (along with the Windows server, remote desktop stuff, phone, fax and Windows firewall and other startup-slowdown trash that I really don't need). I found it rather useless anyway for most of the time.

So... yeah. I can't really help if you want to keep with Symantec, but I do suggest you to never pay for another year's license for Symantec. Nor any other corporate anti-virus software. You can get very effective and sufficient protection for a home PC for free, using freeware programs.
Or, well, perhaps there is something I can do to help you. Go to
Gibson Research Corporation, find the Shields Up! -section, and do some of the security tests they offer there for free. Two that I find extremely good are
-port probe, offering many modes like checking the first odd thousand ports, service ports, specified ports etc.
-firewall tester, which is a small program that you download, start up and starts trying to connect to GRC. If your firewall is worth it's salt, it will intercept the program and ask if it should be allowed, in which case you should answer no (because you want to know if your firewall can stop the program).
These utilities should tell you pretty certainly whether you do have a firewall activated or not. If you have, and if you want to stick with Symantec AV, then you can get rid of the warning prompts by disableing Windows Security Center (Run msconfig, select "Services" -tab, find the Security Center from the list of services, uncheck the box and restart PC.