Hello Brian,
GPG is really what you want to be using for this. OpenSSL is a general toolkit that provide a lot of good functions, but you need to cobble some things together yourself. GPG is meant to handle all of the other parts of dealing with files.
I will expand on what someone else mentioned -- asymmetric encryption is not meant for, and has very poor performance for encrypting data, and also has a lot of limitations. The correct way to handle this is to create a symmetric key and use that to encrypt the data, then use asymmetric encryption to encrypt only the symmetric key.
GPG takes care of this all internally, so that's what you should be using.
Do you have any resources that show examples of how to use GPG internally that takes care of symmetric/asymmetric encryption?
Thanks, - xinhuan