The answer is no, and this can be proved by forcing. In Cohen's model of $ZFC+\neg CH$, he forced over a model of GCH to add $\omega_2$ many Cohen reals, and the result was $2^\omega=2^{\omega_1}=\omega_2$, which would be a counterexample to what you are asking.
It is also a consequence of Martin's Axiom (plus $\neg CH$) that $2^\kappa=2^\omega$ for all $\kappa\lt\frak{c}$.