If you press tab in game you can see how many gold medals you have, and if you look next to the Deaths counter there is an empty circle for a medal there, but there are no medals for Deaths.

Why not? I think it would help if people had more incentive to not die, chaining deaths is probably the most common mistake in Overwatch. I know it might encourage camping/playing too safe, but in quick play I think it will help more than it would hurt, and in competitive good teams wont care about medals.

Gold being the least amount of deaths, obviously, not the most.

We used to give a medal for fewest deaths. We ran into a few issues with doing it. The first was confusion. Players were consistently confused as to whether the medal was for most or fewest deaths. I believe this problem was easily solvable. The second issue had to do with weird behaviors that some players felt compelled to do based on their desire to receive the medal. The medal was ok if it just served as a reflection of what players were already doing. But doing weird things (like hiding back at the spawn room) to get the medal drove too much bad behavior so we removed it.