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".
He added: "I feel very grateful to have had the baton handed over and I'm going to try to do my best."
Showrunner Russell T Davies said Gatwa had impressed him in a "blazing" audition.
"It was our last audition. It was our very last one," the writer and producer said. "We thought we had someone, and then in he came and stole it.
"I'm properly, properly thrilled. It's going to be a blazing future."
The Queer As Folk and It's A Sin creator, who is returning to Doctor Who after departing in 2009, posted a selfie with Gatwa on the Baftas red carpet.