Hello Mdamdedo,
When you are talking about vSphere, what specific product are you referring to? This is because vSphere is the name of the entire Suite of VMware products on which vCenter and ESXi are part of, but there are also other components. So when you say "Does vSphere support remote desktop connection?" is it a bit confusing :smileyhappy:
To start with something that might help:
1. If you have a vCenter Server installed on a Windows Server machine, you could use RDP to login to the Windows OS and then manage vCenter from it (either from vSphere Web Client, HTML5 Client or thick Client if you are using 5.x or 6.0).
2. If you have a vCenter Server Appliance (VCSA), it only supports connections through SSH, Web Client or the DCUI (Accessible as a Console view in the Web Clients).
3. ESXi does not support connections through RDP, only through SSH, via Web Client or DCUI (normally accessible from iLO/iDRAC, etc).
If the above does not clarify your issues, then I would appreciate if you answer my first question so I can provide you with a more exact answer.
Kind regards,
Pablo