My professor told me:
Why is this true? Is there a proof that confirms this?
I was/am assuming that for public key encryption, COA means "other than the public key, ciphertext only". Otherwise, any secure symmetric cipher with the key published becomes a "COA resistant" PKE scheme. With that in mind, access to an encryption oracle cannot possibly help an attacker, since the attacker can already encrypt any plaintext using the public key. $\:$ Therefore, COA resistance implies CPA resistance. $\:$ (for public-key encryption, with my assumption on the definition of COA)
I have a definition of COA-security in my head but I cannot find this definition (applied to public key cryptography) in the literature or reference books.
Under it, textbook RSA is an example of a public key cryptosystem that is COA-secure but not CPA-secure. To be CPA-secure, a necessary but not sufficient property is that encrypting the same message twice does not give you the same ciphertext both times. For this, the encryption function should be randomized. RSA with OAEP is CPA-secure as is Elgamal. To be COA-secure, you basically only need to ensure the private key is not easily computed from the public key.
In any event, the question really comes down to definitions and without providing a definition of COA-security for PKC, it was rash of me to say the statement is true or false (the latter, I did initially, before I edited this answer).
If the only difference between a COA and CPA security game is whether the adversary has access to an encryption oracle, then providing the public key to the adversary gives the adversary all the powers on an encryption oracle. In this case, COA and CPA are equivalent.
However the challenge in the CPA security game based on the adversary choosing a pair of messages for encryption, being given an encryption of one of them (selected at random), and being challenged to state which one. I don't believe the challenge in a COA-secure game for PKC would be the same, as the challenge is premised on the adversary choosing (or at least knowing) the two potential plaintexts candidates.
If the challenges are different in COA and CPA, they are not necessary equivalent.