The public domain translation reverses the Bible quote to be “Love is God,” for cute rhetorical reasons. But that wording was the translator’s choice, not Augustine’s choice. Take a look the other way around, and I’ll prove that the argument is still there.
Augustine, In Epistolam Joannis ad Parthos, Tractatus 9, 10:
“If anyone say,” [John] says, “”I love God,” and hates his brother, he is a liar.” (1 Jn. 4:20)
— So from where do you prove “he is a liar”?
Listen. “For how can he who does not love his brother whom he sees, love God, whom he does not see?” (1 Jn. 4:20)
— So what? Does he who loves his brother love God, too?
It is necessary that he should love God; it is necessary that he should love Love Himself. Can he love his brother and not love love? [No.] It is necessary that he should love love.
— So what? Does he who loves love therefore love God?
For that reason, certainly. Loving love, he loves God. Can you have forgotten what you said a little while back? “God is love.” (1 Jn. 4:16) If “God is love,” anyone who loves love, loves God. Therefore love your brother and be untroubled.
You cannot say, “I love my brother but I don’t love God.” How you lie if you say, “I love God but I don’t love my brother”! So you are deceived when you say, “I love my brother,” if you suppose that you don’t love God.
It is necessary that you who love your brother should love Love Himself. For “God is love;” it is necessary therefore that anyone who loves his brother should love God. For if you do not love the brother whom you see, how can you love the God whom you do not see?
Why does he “not see” God? Because he does not have Love Himself. Therefore, because he does not have love, “he does not see” God. Therefore, because he does not love his brother, He does not have love. Well, therefore “he does not see” God because he does not have love. For if he had love, he would see God, because “God is love,” and his eye would be purged more and more by love, so that he would see that Unchangeable Substance for Whose presence he would always rejoice, which, together with the angels, he would enjoy for eternity.