No. SSL does not work that way - data sent over an SSL connection is encrypted using a symmetric cryptographic algorithm (usually AES, RC4, or similar), using a key that is established during the initial SSL handshake. The public/private key of the server are only used during the initial handshake; after that, they are not used.
Anyways, storing the password this way does not make it any more secure. If it's stored in an encrypted form that can be sent to the server, anyone who managed to get it would be able to use it that way; that encrypted form is password-equivalent, so it's no better than just storing the password!