Alternatively, you could place the CA certificate of the party that issued the certificate for your target web site in the HTTPS_CA_DIR. Most commercial CA's link to it on their web site or you may already have it.
As to the setting for HTTPS_CA_DIR: you could create any convenient directory in your code base to hold the certificate and point to that. You may as well keep it close to your code.
Windows places its certificates in special certificate stores (which explains your result in Internet Explorer) that AFAIK are not that easily accessible to non-MS applications.