Let $C$ be a Jordan curve in $\mathbb{R}^2.$ Then say $U$ is its interior. My problem is if there always exists a metric on $\mathbb{R}^2$ with respect to which $U$ is exactly an open ball.
If $U$ is a star domain, then I think I can set a metric to accomplish it, but for the general cases I'm not sure (I guess it's not correct in general but I haven't come up with a counterexample).
By the Schonflies theorem, for any Jordan Curve $C$ there is a homeomorphism $h$ carrying the pair $(\mathbb R^2,C)$ to $(\mathbb R^2,S^1)$, where $S^1$ is the standard circle. Let $d$ be the usual metric on the plane. Then $d'(x,y)=d(h(x),h(y))$ is a metric on $\mathbb R^2$ that does the job you seek.