Objection 1. It seems that hatred does not arise from envy. For envy is sorrow for another's good. Now hatred does not arise from sorrow, for, on the contrary, we grieve for the presence of the evil we hate. Therefore hatred does not arise from envy.
Objection 2. Further, hatred is opposed to love. Now love of our neighbor is referred to our love of God, as stated above (25, 1; 26, 2). Therefore hatred of our neighbor is referred to our hatred of God. But hatred of God does not arise from envy, for we do not envy those who are very far removed from us, but rather those who seem to be near us, as the Philosopher states (Rhet. ii). Therefore hatred does not arise from envy.
Objection 3. Further, to one effect there is one cause. Now hatred is caused by anger, for Augustine says in his Rule that "anger grows into hatred." Therefore hatred does not arise from envy.
On the contrary, Gregory says (Moral. xxxi, 45) that "out of envy cometh hatred."
I answer that, As stated above (Article 5), hatred of his neighbor is a man's last step in the path of sin, because it is opposed to the love which he naturally has for his neighbor. Now if a man declines from that which is natural, it is because he intends to avoid that which is naturally an object to be shunned. Now every animal naturally avoids sorrow, just as it desires pleasure, as the Philosopher states (Ethic. vii, x). Accordingly just as love arises from pleasure, so does hatred arise from sorrow. For just as we are moved to love whatever gives us pleasure, in as much as for that very reason it assumes the aspect of good; so we are moved to hate whatever displeases us, in so far as for this very reason it assumes the aspect of evil. Wherefore, since envy is sorrow for our neighbor's good, it follows that our neighbor's good becomes hateful to us, so that "out of envy cometh hatred."
Reply to Objection 1. Since the appetitive power, like the apprehensive power, reflects on its own acts, it follows that there is a kind of circular movement in the actions of the appetitive power. And so according to the first forward course of the appetitive movement, love gives rise to desire, whence follows pleasure when one has obtained what one desired. And since the very fact of taking pleasure in the good one loves is a kind of good, it follows that pleasure causes love. And in the same way sorrow causes hatred.
Reply to Objection 2. Love and hatred are essentially different, for the object of love is good, which flows from God to creatures, wherefore love is due to God in the first place, and to our neighbor afterwards. On the other hand, hatred is of evil, which has no place in God Himself, but only in His effects, for which reason it has been stated above (Article 1), that God is not an object of hatred, except in so far as He is considered in relation to His effects, and consequently hatred is directed to our neighbor before being directed to God. Therefore, since envy of our neighbor is the mother of hatred of our neighbor, it becomes, in consequence, the cause of hatred towards God.
Reply to Objection 3. Nothing prevents a thing arising from various causes in various respects, and accordingly hatred may arise both from anger and from envy. However it arises more directly from envy, which looks upon the very good of our neighbor as displeasing and therefore hateful, whereas hatred arises from anger by way of increase. For at first, through anger, we desire our neighbor's evil according to a certain measure, that is in so far as that evil has the aspect of vengeance: but afterwards, through the continuance of anger, man goes so far as absolutely to desire his neighbor's evil, which desire is part of hatred. Wherefore it is evident that hatred is caused by envy formally as regards the aspect of the object, but dispositively by anger.
Source: Summa Theologica (New Advent)