I know that if $X,Y$ are jointly normal with unit variances, then $E[X\vert Y=y] = \rho y$, which is of the form in the title (with $\rho$ being the covariance). Also, I think as long as the means are $0$ then the conditional expectation will have the proper form.
However, what I don't know is whether the other direction is true. That is, I don't know whether $E[X\vert Y=y] =\gamma y \implies $ that $X,Y$ are jointly normal.
I feel like this isn't true, but I can't prove that it isn't (I don't even know how I'd approach it).
I'm not looking for a proof, although I'm okay if one is posted.
Answer is no.
Carmichael561 gave a simple counterexample.
"If $X=\gamma Y$ then $E[X\vert Y=y] = \gamma y$, regardless of the distribution of $Y$"