No, In Identity Platform Authentication Using OpenID Connect with Apple is not possible with Private Key for that You have to Generate Client Secret.
The reason why we need to create Client secret(for Apple Provider) is mentioned here in the Article as :
Apple’s support for OAuth2 has a number of differences compared to all
the other social providers. Most providers (Google, Github, Facebook)
generate the client_id and client_secret for you. You just need to
keep them in a secure place on the server. In the Apple process, you
first download a private key, and then generate a client_secret using
that private key. The other difference with Apple is that they don’t
support using localhost as a redirect. In the end, I initiate the
OAuth2 process on the Flutter client; but all the redirects and the
token exchange process happen on the server.
In order to generate the client secret, You can follow the steps mentioned in the Article.
For more information, you can refer to the Answer on how to generate client secrets on Apple Platform.