Is it possible to use a wildcard SSL cert on an ASA? That is, instead of getting a specific cert with the FQDN of the ASA, we would use the wildcard cert issued?
Absolutely, it's especially needed in ASA vpn load balancing environments. When you connect to a FQDN that translates to a load balancing IP, one of the ASAs will do an http redirect to its individual hostname, your browser (or AnyConnect) will attempt that connection and ASA needs to have a certificate for that specific hostname. Having a wildcard cert on all ASAs resolves this. I've got this running on several customers.
If you need help with configuration, let me know.
You can either generate private keys on the ASA (and later export it to another ASA or other non-cisco devices), or you could import an existing wildcard certificate with the private keys (in PKCS12-BASE64 format)