According to the problem, the hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) has an eccentricity of $2$,
thus $e= \dfrac{c}{a}=2$, which means $c=2a$,
and since the distance from the right focus to one of the asymptotes is $\sqrt{3}$, we have $b= \sqrt{3}$,
also, from $c^{2}=a^{2}+b^{2}$, we get $4a^{2}=a^{2}+3$,
which leads to $a^{2}=1$,
therefore, the equation of the hyperbola is: $x^{2}- \dfrac{y^{2}}{3}=1$;
hence, the correct choice is: $\boxed{B}$.
According to the problem, by using the formula for the eccentricity of a hyperbola, we get $e= \dfrac{c}{a}=2$, which means $c=2a$. Also, from the properties of the hyperbola, we find $b= \sqrt{3}$. Combining this with $c^{2}=a^{2}+b^{2}$, we can calculate the value of $a^{2}$, and substituting the values of $a^{2}$ and $b^{2}$ into the equation of the hyperbola gives us the answer.
This problem examines the geometric properties of the hyperbola, noting that the distance from the focus to the asymptote of the hyperbola is $b$.