The answer is no, even if you consider only injective functions. Consider any infinite cardinal $\kappa$, and let $X=\omega\times\kappa$ be the disjoint union of $\kappa$ many copies of $\omega$. Let $f:X\to X$ be the function that shifts within each copy separately. So $f$ has no fixed points. But meanwhile, for any subset $A$ of the copies of $\omega$, let $g_A$ be the function that shifts within the copies of $A$, but is the identity on the opices of $\omega$ not in $A$. So every $g_A$ commutes with $f$, and there are $2^\kappa$ many such $g_A$. So this is a counterexample to the converse implication, since $|\text{Com}(f)|=2^\kappa>\kappa$ but $2^{|\text{fix}(f)|}=1<\kappa$.

The answer is no, even if you consider only injective functions. 

Consider any infinite cardinal $\kappa$, and let $X=\omega\times\kappa$ be the disjoint union of $\kappa$ many copies of $\omega$. Let $f:X\to X$ be the function that shifts within each copy separately. So $f$ has no fixed points. But meanwhile, for any subset $A$ of the copies of $\omega$, let $g_A$ be the function that shifts within the copies of $A$, but is the identity on the copies of $\omega$ not in $A$. So every $g_A$ commutes with $f$, and there are $2^\kappa$ many such $g_{A}$. So this is a counterexample to the converse implication, since $|\text{Com}(f)|=2^{\kappa}>\kappa$ but $2^{|\text{fix}(f)|}=1<\kappa$.

The answer is no. Consider any infinite cardinal $\kappa$, and let $X=\omega\times\kappa$ be the disjoint union of $\kappa$ many copies of $\omega$. Let $f:X\to X$ be the function that shifts within each copy separately. So $f$ has no fixed points. But meanwhile, for any subset $A$ of the copies, let $g_A$ be the function that shifts within the copies of $A$, but is the identity otherwise. So every $g_A$ commutes with $f$, and there are $2^\kappa$ many such $g_A$. So this is a counterexample to the converse implication, since $|\text{Com}(f)|=2^\kappa>\kappa$ but $2^{|\text{fix}(f)|}=1<\kappa$.

The answer is no, even if you consider only injective functions. Consider any infinite cardinal $\kappa$, and let $X=\omega\times\kappa$ be the disjoint union of $\kappa$ many copies of $\omega$. Let $f:X\to X$ be the function that shifts within each copy separately. So $f$ has no fixed points. But meanwhile, for any subset $A$ of the copies of $\omega$, let $g_A$ be the function that shifts within the copies of $A$, but is the identity on the opices of $\omega$ not in $A$. So every $g_A$ commutes with $f$, and there are $2^\kappa$ many such $g_A$. So this is a counterexample to the converse implication, since $|\text{Com}(f)|=2^\kappa>\kappa$ but $2^{|\text{fix}(f)|}=1<\kappa$.