See the proof of Theorem 2.15.6 from the Handbook of categorical algebra, Volume 1, by Borceux. The result is stated for $\mathcal C$ small, but the proof also works for large ones (since he proves directly that $F$ satisfies the universal property, rather then assuming that the colimit exists and proving that the comparison map is an iso).

Simon makes a good point. In my head the question is asking whether the class of all weighted colimits is the "saturation" of the class of conical colimits (in the sense of Kelly-Schmitt). That is, whether every $\mathcal V$-category with conical colimits also has all weighted colimits, and every $\mathcal V$-functor preserving conical colimits, also preserves all weighted ones. The answer is negative for $\mathcal V=\mathbf{Pos}$ since discrete posets form a $\mathbf{Pos}$-category that has all conical colimits but lacks the weighted ones.