No, you cannot do it (regretfully IMO). Acrobat JavaScript API has only support for certificate encryption but not for password encryption. One can create a C++ plug-in with a security handler, but Acrobat/Reader currently supports only signature part of the Security Handler but not the encryption part (by design a Security Handler can handle both digital signatures and encryption but Acrobat/Reader recognizes only the digital signature part in the 3rd-party security handlers).
I was told that in the past US government imposed export restrictions on software containing encryption and that that was the reason for this Acrobat behavior. I do not think that these government export restrictions are still in place but Acrobat has not changed in this regard. I guess there were just not many important customers who asked for it.