In my experience, you will need a trusted certificate on the web adaptor server as well. Since the web adaptor directs traffic to the GIS Server, the web adaptor needs the trusted cert. I have trusted certs on both the web adaptor and GIS server.
In case you need it, here's how I get up the 443 binding in IIS on the web adaptor server:
1. Open IIS 7.5. Expand to see Sites.
2. You should see the Default Web Site. Select it.
3. Under Actions on the right, click Bindings...
4. Add a binding for type https on port 443.
5. Select a SSL certificate (which should already be loaded to the server by the Network gurus.) If it isn't, they will need to create the certificate and import it onto the server. Sorry, I can't explain this process more as it is all a bit voodoo to me.
6. Click OK.
7. (for our setup, you may not need). With the website selected, double click on Authentication under IIS. Confirm that Anonymous authentication is enabled and everything else is disabled.