Salih, who served as the prime minister of Iraqi Kurdistan in 2009-2012, was elected by the country's parliament earlier in the day. Salih, backed by the Patriotic Union of Kurdistan (PUK), won in the second round of voting.
The parliament was formed after May 12 general election.
Abdul-Mahdi was the minister of oil in 2014-2016, under Prime Minister Haider Abadi, Salih's predecessor.
Previously, the Patriotic Union of Kurdistan (PUK) nominated former Iraqi Kurdistan's leader Barham Salih for the post of Iraqi president. The Patriotic Union of Kurdistan has already nominated Salih for the presidency four years ago but the office was taken by Fuad Masum.