Here is a response from GoDaddy tech support regarding this issue:

Thank you for contacting Online Support. You can use CURL to make a connection of the kind that you require. On our shared hosting, any CURL applications that make secure HTTP connections (HTTPS over port 443) need to pass through a proxy server. The address for this proxy server is proxy.shr.secureserver.net and connections must specify the use of port 3128. While we cannot provide custom scripting support, you can reference an example CURL script though the Help Center article below:

• Using CURL to Connect Using HTTPS (http://help.godaddy.com/article/289)

Please let us know if we can help you in any other way.