Since he's named after his father, he should be Kenyatta Watson, Jr. Kenyatta Watson, II would indicate that he has the same name as a family member other than his father, for example his brother or uncle.
Another rule people seem to miss with postnomials is that they change after someone dies. For example if Kenyatta Watson, Jr. has a son and gives him the same name, he would be Kenyatta Watson, III. If Kenyatta Watson, Sr. then dies, Kenyatta Watson, Jr. becomes Kenyatta Watson, Sr. and Kenyatta Watson, III becomes Kenyatta Watson, Jr.
I'll preemptively do this: