Wireguard is not PGP. In particular, Wireguard doesn't do several things that PGP does, because doing those things is very difficult to do securely. One of those things is encrypting anything with a public key: Wireguard only does elliptic-curve Diffie-Hellman key agreement.
In general, you should never encrypt a message with asymmetric cryptography (public/private keys) directly. You should either use a Key Encapsulation Mode (KEM), or a Diffie-Hellman style key agreement exchange (probably elliptic-curve based). You then use the resulting shared secret key to encrypt messages using an Authenticated Encryption with Associated Data (AEAD) symmetric cipher, like AES-GCM or ChaCha20-Poly1305 (Wireguard does this).
Asymmetric cryptography is good for key exchange, either with a KEM or a DH agreement. It's also good for signing and verification of messages. It's not good for encryption.