Let $C$ be a category and assume either that $C$ has all binary pullbacks or that $C$ satisfies right calculus of fractions. In both cases the localization of $C$ at every morphism (i.e. the groupoidification) can be represented by spans, i.e the objects are the same as in $C$ and morphisms are spans $A\leftarrow B\rightarrow C$ modulo some relation (see the link, section Construction of the localization).

My question is now: Does having binary pullbacks imply the right calculus of fractions? Or vice versa? Or is there no general relation between the two?