The 29-year-old will become the 14th Time Lord on the popular science fiction show, and the first person of colour to play the lead role.
Scottish actor Gatwa, who was born in Rwanda, is best known for starring in Netflix's sitcom Sex Education.
He told BBC News: "It feels really amazing. It's a true honour. This role is an institution and it's so iconic."
Speaking on the red carpet before Sunday's Bafta TV Awards, where he was nominated for Sex Education, Gatwa said the role of the Doctor "means a lot to so many people, including myself".