  Atomic Fermi gases provide an ideal platform for studying the pairing and
superfluid physics, using a Feshbach resonance between closed channel molecular
states and open channel scattering states. Of particular interest is the
strongly interacting regime. We show that the closed-channel fraction $Z_{cc}$
provides an effective probe for the important many-body interacting effects,
especially through its density dependence, which is absent from two-body
theoretical predictions. Here we measure $Z_{cc}$ as a function of interaction
strength and the Fermi temperature $T_\text{F}$ in a trapped $^6$Li superfluid
throughout the entire BCS--BEC crossover, in quantitative agreement with theory
when important thermal contributions outside the superfluid core are taken into
account. Away from the deep BEC regime, the fraction $Z_{cc}$ is sensitive to
$T_\text{F}$. In particular, our data show $Z_{cc} \propto T_\text{F}^{\alpha}$
with $\alpha=1/2$ at unitarity, in quantitative agreement with calculations of
a two-channel pairing fluctuation theory, and $\alpha$ increases rapidly into
the BCS regime, reflecting many-body interaction effects as predicted.
