When rolling a 20 for a Death Saving Throw, can you move and take Actions that turn?

My intrepid adventuring character has been attacked, and the damage reduced him to 0 hit points and falls unconscious. On his next turn, he rolls a Death Saving Throw, and gets everybody’s favorite die roll: a natural 20! He regains 1 hit point, and thus regains consciousness.

Now that he’s conscious, can he still move and take an action for his turn? The rules say that you roll a Death Saving Throw “Whenever you start your turn with 0 hit points,” but it’s not clear to me if that’s replacing one’s normal turn, or if it’s just at the start of one’s turn, and the only reason one normally doesn’t follow it with a movement and action is because unconscious creatures are incapacitated and can’t move.

There’s some discussion in the comments of an answer to “Do you regain consciousness immediately on a 20?”, but it doesn’t seem to have an authoritative referenced resolution to the discussion, and it seems that if it’s worth talking about in comments it’s worth asking as a full-fledged question.