Based on the posted scenario, he stopped at first, and then stole while they weren't paying attention (the ball was retrieved and returned to the pitcher, you don't have to wait for the pitcher to be on the mound to get a steal). To be a double, you can't stop at a base (I know we could come up with examples of where a ball is hit in the corner and a runner stops then relies the ball hasn't been fielded yet, so he advances but I don't think this is the case). Since there are no mental errors and he didn't advance on the throwing error, it has to be a steal to account for his advancement.
If your interpretation is that a play has to made, then any runner who advances on a traditional steal and no throw down is made because it is to late, then what would you score it.
10.07 Stolen Bases And Caught Stealing
The official scorer shall credit a stolen base to a runner whenever the runner advances one base unaided by a hit, a putout, an error, a force-out, a fielder's choice, a passed ball, a wild pitch or a balk, subject to the following: