Technically yes, but practically no. Even if you stop now, you probably have used Google services in the past and they already have those info. Even if you do not use Google apps, there are lots of way for your phone to connect to Google server (various core Google apks). Lots of apps and websites use Google Analytic as well, although that info is supposed to be anonymous, or apps that utilize Google services in some way or purchased apps that required verification. You basically have to micromanage the permission of every app and this isn't exactly easy and is time consuming, slips up are easy too.
My recommends are:
Most effective: Be lke Richard Stallman, stop using internet completely, except for devices that isn't your. Don't use telephone at all. Have no WiFi running in your house and remove your address from Google Maps, then move to South Georgia and the South Sandwich Islands.
Less effective: Don't use 'gapps' and Google apps at all. You can use Amazon AppStore
to download apps. Or use Firefox OS or Ubuntu Touch instead.
Less effective alternative: Use permission management. There are many ways to do this. Android 4.3 have AppOps and there are shortcut apps
to let you access to the hidden menu, however it's not very convenient to use. It's better to use CM or SlimRoms which in addition have the Privacy Guard toggle feature (Settings -> Security-> bottom of list). SlimRoms let you turn it on for installed apps by default. It blocks access to contacts and location. I'm not sure if it block network and phone IDs info though.
For more refined permission management, there is the Xposed Framework module called XPrivacy
. It has a high learning curve and blocking the wrong permission will cripple your apps or make it not working. There is also LBE Security Master and OpenPDroid, but I much prefer XPrivacy.
You can also use AFwall+
which modify the Linux kernel's iptables to block internet connection, this is the most assured method to block internet connection access for apps imo.
If you still need to make use of Google apps or services, such as facelock and photosphere, but want to avoid installing 'Google services framework', you can use these gapps packages
and not install the core package. Keep in mind, 'Google services framework' is important for connecting to Google server and without it some 'find your phone' apps won't work like Cerberus.
You can see if your phone is connected or syncing to Google server by the color of the WiFi icon color, gray means no, and blue means yes.