I have an EC2 server set up inside a VPC on Amazon. The server is reachable from the outside via its elastic ip address; however, when I attempt to ping that ip from inside the remote desktop session, the server cannot reach itself. Is there a way to make this happen?
It should work.
As much I understand, you haven't allowed the ping requests to be responded by the Machine in your Firewall rules. So, you shouldn't be able to ping the IP from outside too.
To do the same, Create a rule in your Security group by following steps.
After this, you should be able to ping the Elastic IP from anywhere.