Should I push my child into sports?

It is important for you as a parent to remember that exposing children to new ideas, activities is about the child learning and exploring his or her own skills and interests and not the needs or wishes of the parent or what the parent wants for the child.
Encouragement try new things and understand commitment and responsibility when they join a group is important but it is also very important to talk with your child and truly understand your child's thoughts and feelings.  There is not one good answer regarding how to respond or  react to child not wanting to do something.  There are many factors to be considered and every child is unique.  Their thoughts and experiences are unique also.  (physical issues, emotional issues, social issues, maturity levels, unpleasant experiences, etc...) better yet is the process for resolving the issue at hand...this is what will determine your child's future receptiveness and responsiveness to new experiences and this is the real teaching opportunity.