The "Unable to establish a connection to www.arcgis.com." error message indicates a connectivity problem between ArcGIS Pro and arcgis.com. It is quite possible that something else happened on the first machine around the same time you installed Pro on the second. I can't think of a direct connection between the second install and the connectivity failure.
I have seen the issue before after setting up the Fiddler web debugging utility or anything else that causes a Proxy server to get set for your LAN connection. If the proxy is configured incorrectly, it can interrupt your connection with arcgis.com. Check two things:
On the machine with the problem:
- See if you can connect to www.arcgis.com using a web browser. If it fails in the browser, it's specific to the machine configuration and not isolated to ArcGIS Pro.
- Check the value of the Proxy server checkbox: Internet Settings control panel >> Connections tab >> LAN settings >> Proxy server section "Use a proxy server for your LAN". If it's checked and you are able to uncheck it, try unchecking it an running ArcGIS Pro again.
Tushar Jadhav reminded me of the troubleshooting tips.