Mac Pro 2012 runninig Mountain Lion not connecting to internet

Using 1.1.1.1 is a diagnostic. it fixes NOTHING.

If you try it and get the colorful CloudFlare splash page on your screen, your internet connection is WORKING, and you can deal with the much narrower problem of getting a good DNS server active.

if you can not connect to an all-numeric IP address, something more fundamental than bad/missing DNS is hindering your internet connection.

what results did YOU get when you entered 1.1.1.1 ??

your DNS is being provided ONLY by your Router at address 192.168.0.1. the problem is that your Router does not know the IP addresses of a lot of web sites, and you have no backup DNS numbers entered.

I suggest you manually enter the numbers for OpenDND:

208.67.220.222

208.67.222.222

if your internet connections then allow you to FIND a few websites, that's good. But you won't be able to connect to most of them. You need to install El Capitan.

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In 2015, researchers discovered that Secure Socket Layer (SSL) Internet encryption was not nearly as secure as was thought, and needed to be replaced. Internet encryption quickly moved to Transport Layer Security (TLS) which was rapidly deployed across the Internet. Over time, sites tightened requirements for what was acceptable for encryption.

Apple issued Safari version 9 in 10.11 El Capitan version of MacOS, which included TLS encryption. It was later provided by Security Update into the two previous versions of MacOS. 10.10 Yosemite and 10.9 Mavericks, PROVIDED you applied all available software updates.

MacOS versions older than those can not make a lot of secure [httpS:] connections on the Internet, because by todays standards, your proffered SSL encryption is considered inadequate.

How to download and install macOS - Apple Support

I had the same problem with OS X Lion on a Mac Mini mid2011 (only the settings app connected to the internet, Apple ID cannot be loaded and no App Store; no Safari connecting to the internet). This problem gone out by updating to High Sierra. Try updating to the latest MacOS version available by recovery. (To boot into the latest available recovery via internet press: Command - Option - R or *if you use a Windows keyboard* Windows - Alt - R)

Launch a browser and enter this easy to remember all-numeric IP address:

1.1.1.1 

in the address bar,  and press return.

If you get the CloudFlare** splash page, your internet is working, but DNS lookup may not be working.

**CloudFlare is NOT part of the solution, they just own the easiest to remember all-numeric IP Address.

<<. When I type 1.1.1 I get a screen listing links containing 1.1.1.1. >>

what software is producing that list? you do not want wikipedia to explain it to you -- you want that EXACT website address.

try entering http://1.1.1.1

What EXACT symptoms do you have that lead you to say, "It will not connect to the internet via WiFI or Ethernet"

I don't think this will work. I have tried also but you can just see if you are connected to the internet.

there are ways you can use links Inside that error message to (possibly) show the certificate and then tell it to

'connect anyway'