Is it permissible for a pharmacist to give a doctor a gift in exchange for the doctor directing patients to his pharmacy, knowing that the doctor recommends certain medications in exchange for gifts, and that the pharmacist does not intend to infringe upon the rights of patients?

There is no harm in offering a gift to a doctor in exchange for him directing patients to your pharmacy, due to the Prophet's (peace and blessings be upon him) saying: "Whoever does you a favor, then recompense him." This is permissible provided that it does not lead to the squandering of the rights of others.