Is the only reason we exist to worship God? Very plausibly, no. We certainly do exist to worship God, and that is our primary function, but angels do that. No, the Bible teaches that man was made "in the image of God," (Gen. 1:26) and that this image is not physical, but instead in "righteousness and true holiness" (Eph. 4:24), which is to say moral goodness. We have rational faculties, free will, and we were created with moral innocence. We not only were created to worship God, but we were created to love God, and to be with him in fellowship (close relationship); we were also created for God to love us! Several of these concepts will surely be interrelated, but these are identifiable biblically, theologically, and philosophically as purposes of God with respect to creating humans. So, no, we do not have only one purpose, though each of our sub-purposes is plausibly worship of God (though not identical to it).
I sense another question coming on the heels of this, though, that probably led to his discussion of evil. That question is, "Is God worthy of worship at all?" That question can be answered by appealing to what is known as Perfect Being Theology (PBT). PBT states that God is the greatest conceivable being, he is the most perfect being there is, or could ever be (even in principle). As such, he has perfect attributes, such as moral perfection as the standard for morality, holiness, righteousness, omnipotence, omniscience, etc. These attributes make him worthy of worship since it is not the case that God simply conforms to all of the objective moral norms; on Christian theology, rather, he just is the foundation of these objective moral norms. We could not worship someone who simply met the standard (although that is morally praiseworthy); we can only worship someone who is that standard.
Now he will want to say that the Christian God doesn't match up, but unless he has something new to say with respect to the question of man's purpose and God's worship, he has to grant, at least in principle, that this objection is overcome.