An air pump is very unlikely to have killed your fish. There are several instances where air pumps are used for flow (air lifts) however in a smaller tank, it might prove problematic with salt creep and the like.
So, as an answer directly to the questions.
1. No, you do not need an air pump for a successful, sustainable reef aquarium.
2. No, it is very unlikely that having (or not having) an air pump was the cause of your fish's death.
Flow is best provided by powerheads in these smaller tanks (and even large display tanks, I primarily see air-lifts used on large frag tanks/vats).
Theres a multitude of things that could have resulted in the death of your fish. If you can, post the water parameters (temp, pH, SG, Ammonia, Nitrite, Nitrate, and whatever else you have test kits for). If you don't have any test kits, make sure to bring a water sample to your LFS and ask them if they could test it for you. Also, the fish could have perished from stress. What kind of fish was it?