You don't need root to connect to a Google account. That only reflects poor connectivity. On what basis do you believe that root has any bearing on your difficulties? You didn't give us any details of that or anything else which is unfortunate and precludes narrowing anything down. But it would be something like ISP /networking problems, a restrictive firewall or possibly a corrupt cache or account settings on your phone.
Try wiping your partition cache from recovery mode. If the problem persists afterward, then try deleting your Google account on your phone and then adding it again. You should also try connecting from another internet connection to try and discern if connectivity issues with your ISP are at issue.
To effectively troubleshoot this though you need to provide a lot more details. Verbatim error messages, details and screen shots of your network connectivity, observed patterns e.g. do other apps have issues or work fine, is the problem sporadic or continuous and anything else even remotely pertinent for a start.