Given the formula $\Delta G = \Delta H - T\Delta S$. It is commonly written that all spontaneous reactions necessarily have $\Delta G < 0$. I was curious if the converse is true: Are all reactions with $\Delta G < 0$ spontaneous. If not, why?
No, Gibbs energy is only the relevant state function in the special case of constant temperature and pressure.
In the wording of IUPAC:
partial derivative of Gibbs energy with respect to extent of reaction at constant pressure and temperature [is negative] for spontaneous reactions
In the special case of constant temperature and volume, Helmholtz energy is the relevant state function.
Most generally, a process is spontaneous if the entropy of the universe (system + surroundings) increases.