I suppose you want uniqueness assuming some initial condition. 

Even when $F$ is continuous and $X$ is $C^1$ you don't have uniqueness. For $t\geq 0$ $X(t)=0$ and $X(t)=t^2$ are both solution of $X'=2\sqrt{X}$.

Maybe with stronger assumptions on $F$ you can do something but I'm not sure.