The arrest warrant was issued by the investigating magistrate of the Vatican City State Tribunal. Msgr Capella, who was in office until last year, was recalled from Washington last September by the Vatican.
It is the latest case the Catholic Church has faced after repeated criticism for the way it has handled scandals over paedophile priests.
The US State Department notified the Vatican in August through diplomatic channels of a possible violation of child pornography laws by a member of the diplomatic corps of the Holy See in Washington, the Vatican said in September.
The United States made "an official request" for the man's diplomatic immunity to be lifted but the Vatican refused, said a US official on condition of anonymity.
In February, the Vatican said it had renewed its anti-paedophile panel in a bid by Pope Francis to quell the global scandal over the sexual abuse of children by priests.