Here are two counterexamples:
(1) the set defined by $xy=1$, $|x|=1$,
(2) the set defined by $y^2=x^3-x$, $|x|=2$.
In both cases a compact Riemann surface bounded by the given curve would have to lie in the solution set of the given polynomial equation. In (1) this is impossible. In (2) it's possible but not with a disk.
EDIT Or an ordinary round circle in $\mathbb R^2\subset \mathbb C^2$. This is the same as (1) up to a complex-linear change of coordinates.