Former astronaut Mark Kelly said Tuesday he will run for U.S. Senate, challenging Arizona Republican Sen. Martha McSally, in what Democrats view as one of the most vulnerable GOP Senate seats in 2020.

Mr. Kelly, 54 years old, is a former American astronaut and a retired U.S. Navy captain. He is the husband of former Democratic Rep. Gabrielle Giffords, who was shot in 2011. After the shooting, he and his wife became political activists, helping candidates and supporting gun-control legislation.

In...