The appeal was sent in the Pope's name by Vatican Secretary of State Cardinal Angelo Sodano'.
He said it was inspired by "profound human compassion" and by the "firm belief in the dignity of human life and by respect for every person's freedom of conscience and religion."
President George W Bush and several other Western leaders have expressed grave concern at the threatened death penalty for Abdur Rahman.
Sharia (Islamic law), on which Afghan law is partly based, stipulates death for apostasy - the abandonment of one's religious faith.
"I am certain, Mr President, that the dropping of the case against Mr Rahman would bestow great honour upon the Afghan people and would raise a chorus of admiration in the international community," the letter said.
The Afghan constitution says "no law can be contrary to the sacred religion of Islam" but also says it will abide by the Universal Declaration of Human Rights, which enshrines freedom of religion.
International pressure on Afghanistan to respect Rahman's religious freedom and release him from jail has been met in Afghanistan by increased calls for him to be tried under Islamic law and executed, and a threat of rebellion if the government frees him.
The controversy threatens to drive a wedge between Afghanistan and the Western backers who secure and finance the country.
Rahman's trial is due to start in a few days.