I had this issue on the vCenter Server Appliance.
I logged into the web interface on port 5480 and changed my DNS servers.
After I saved the changes and rebooted the appliance I got the web client failed to connect error (when accessing the vSphere Web Client on port 9443).
The problem had to do with the web interface on port 5480 incorrectly displaying my hostname (and subsequently saving the incorrect hostname).
I setup the hostname as server.domain.local via the blue console screen when running the script /opt/vmware/share/vami/vami_config_net. I did a certificate replacement procedure and everything had been working fine.
The problem has to do with the web interface on port 5480, it displays the hostname as server instead of server.domain.local.
So when I updated the DNS servers, it saved the hostname as server instead of server.domain.local.
Once I changed the hostname to server.domain.local (in the web interface on port 5480) and rebooted the server appliance, everything worked as per normal, I was able to log onto the vSphere Web Client on port 9443.