Do not depend the old he/who him/whom trick blindly but, in this case, it does actually help.
An Open Letter To Those Whom I Have Hurt
Is your answer and if you substitute third person pronouns you can see why.
If you split up the sentence and just use the last phrase, you can see how who would not be strictly correct. This is because him/whom is the grammatical object and therefore uses the objective pronoun.
I have hurt him [not he]
This then leaves me to answer to greater question: is whom always correct?
As with most questions regarding written or spoken modern English, this is a matter of register and formality. Generally, in more formal uses, whom is most appropriate, however, more casually and more often when spoken, misuse of who is perfectly acceptable.
I prefer to stick to the old ways of using the correct case when possible but public attitude has generally turned towards a multi-purpose who.