From a higher level point of view, the Poincare inequality is about comparing seminorms of functions. For example, it is reasonable to expect something like:
$$\DeclareMathOperator{\dL}{d\!}\int_\Omega ( u - \overline u )^p \dL x \leq C \int_\Omega | D u |^p \dL x$$
Your suggested inequality looks like trying to compare two scalar products, except it has been modified on the right-hand side to be positive.
