I have had this.
Go to the ASA's web page. Install the certificate in your trusted root.
Re try the ASDM and then it works. This is what I did and it worked for me without downgrading Java....
I opened the Java control panel again and I installed the cert to the Secure type area and Signer CA, then the launcher worked fine. The trusted CA wasn't enough.
Be sure to add the hostname or IP address to the sites list in the Security tab as well.
Cisco just told me to downgrade Java, thanks Cisco....