We've got GIS server 10.2.2 running on a centOS machine, and the web adaptor (also 10.2.2) installed on tomcat7 sitting behind apache on the same machine. The interface at 6080 can be accessed both from off-machine and from localhost. However, when trying to configure the web adaptor, I keep getting the error "Please verify that the GIS Server URL can be accessed successfully."
I for one cannot figure out why it can't connect. I've tried using (in the GIS Server URL field) http://localhost:6080, http://localhost:6080/arcgis/, http://[fqdn]:6080, previous /arcgis/, http://127.0.0.1:6080, previous /arcgis/, and even the assigned IP of the machine itself (again, with both variants).
The tomcat logs are not helpful, providing the same error message obtained from the web interface.