Does ECIES imply authenticity? Assume:


Alice and Bob both generate separate EC keypairs
Alice obtains Bob's public key, and together with her private key creates a shared secret key
Alice encrypts a message using the shared key and some mode of authenticated encryption (AES-GCM, for example)
Bob generates the same shared secret key using his private key and Alice's public key
Bob decrypts the message, paying attention to whether or not the auth-tag is valid


My question is; if Bob is confident that Alice's public key really belongs to Alice, then can he be confident that it was Alice who encrypted the message? Or to put it another way, in this scenario does the auth-tag effectively serve as a signature created with a public-key? Intuitively I want to say yes, but I just want to make sure I'm not overlooking anything.
I guess the answer is no, as long as you are using ECIES then this protocol does not work - you cannot trust the public key of Bob, which is required for ECIES. 

You could however use ephemeral-static Diffie-Hellman, using ECDH as cryptographic algorithm. Alice would supply the static part as her public key is trusted, Bob may use any key pair. That means that Bob can trust Alice, but not the other way around. This may however be sufficient for some protocols (TLS normally does not perform client authentication either).

Of course, you may just as well use a static key pair for Alice instead of randomly generating it again and again, requiring to set up trust for each key pair creation.