Is there a way to do single key-pair asymmetric encryption? All of the information on asymmetric encryption I can find uses key-exchanging to get a mutual symmetric key.

What I am trying to do is encrypt a message with a public key and have it only readable by someone with the private key. ​ I don't even want the sender to be able to read it after it's been encrypted. ​ I don't care about the identity of the sender, only that if someone compromises the sender's computer, they can't derive a symmetric key and decrypt what was previously sent.

Is this possible?
Yes, RSA is an example of a cryptosystem where this is possible. The message is encrypted using the recipient's public key only and even the sender could not decrypt it.

However, in the comments you mention that you would like to minimize storage requirements. RSA would require e.g. 2048 bits for just the message. In comparison, with ECIES sending a one-time key with the message you are looking at a minimum of about 512 bits for a one AES block message, 256-bit curve and 128-bit MAC.

So it is not really wasteful to do it like CodesInChaos suggested.