Given that $2x$ is the cube root of $216$, we start by expressing this relationship mathematically:

$$2x = \sqrt[3]{216}$$

We know that $216 = 6^3$, so we can rewrite the equation as:

$$2x = \sqrt[3]{6^3}$$

This simplifies to:

$$2x = 6$$

To find the value of $x$, we divide both sides of the equation by $2$:

$$x = \frac{6}{2}$$

$$x = 3$$

Now, we need to find the square root of $x + 6$. Substituting the value of $x$ we found:

$$x + 6 = 3 + 6$$

$$x + 6 = 9$$

The square root of $9$ is:

$$\sqrt{9} = \pm 3$$

Therefore, the square root of $x + 6$ is $\boxed{\pm 3}$.