No you do not have to have one.
However, my guy wouldn't have done well in only the lap belt at only two, so we used the carseat until he was 3.5 (when I was boarding an almost full plane along with my son and was exhausted, and the gate agent asked one more time if I wanted to gatecheck it and I said yes). Now he does wonderfully, but at 2 he would have been sliding out and unclicking it all flight long...so although it isn't required, I'd highly recommend it.