DFW Controller described a potential conflict between an arrival and departure from Runway 17R due to an unexpected go around event; the reporter's utilization of visual separation questioned.

Narrative

I cleared an A300 for takeoff on Runway 17R (RNAV SID AKUNA). A B737-800 was about 2 mile final for Runway 17R when they stated they were going around. I noticed the B737-800 appeared high and asked to verify they were going around; and they replied affirmative. The A300 was airborne really quickly and climbed very aggressive. The B737-800 was now only about 2.5 miles behind and climbing out as well. I issued visual separation instructions to the B737-800 and he read everything back to me very well. I switched the A300 to departure and turned the B737-800 to a heading of 150. The pilot asked me 'how high do you want me' and I replied maintain 3;000 and that I was going to coordinate a west-side arrival since he was already parking at the West-side. I looked at the B737-800's altitude and he was at 2;700 climbing to 3;000 and the A300 was already outside the upper RADAR set filter limit of 5;000. I asked the B737-800 if they went around due to the high angle of approach; he replied yes and they didn't think they would achieve a stable approach. When I switched the B737-800 to departure there was more than enough vertical separation. While on break after this session; I found out that the B737-800 did not reply with his full call sign upon acknowledgment of the visual separation; thus the rule was not usable. My Supervisor advised me that the 150 heading did not protect enough for the AKUNA SID departure. Because the B737-800 had turned to the 150 heading before even reaching the runway and the aggressive climb rate of the A300; there was supposedly no error involved after all. I was told two different things within an hour about the error's outcome; so I wanted to report this. Recommendation; I shouldn't have pushed the A300 out in front of the B737-800 arriving. I felt I had the time; and in fact did have time to clear the A300 for takeoff in enough time for the B737-800 to land without going around. Last; I should have coordinated a 130 heading with approach and got divergence right off and not worry about the rest of tedious and cumbersome visual separation requirements.