If you take $e=1$, then the condition imposed on $e$ translates to: for every $x \in R,$ either $x$ or $1-x$ is a unit. This will be satisfied in any local ring $R$, while $R=(e)$ will not be minimal unless $R$ is a field. This already shows that the conclusion does not hold in general. 

(If $e=1$ seems too trivial, this example should survive replacing the local ring $R$ by any ring of the form $R\times S$ and $e=1$ by $(1, 0)$).

On the other hand, if the Jacobson radical $J(R)$ of $R$ is $0$ and a nonzero idempotent $e$ has the imposed property, then $(e)$ is indeed minimal:

One needs to check that for $0 \neq x$ divisible by $e$, i.e. $ex=x$, one has $e$ divisible by $x$. In other words, if $x=ex$ is an element not dividing $e$, then $x=0$. The strategy is to show that $x$ lies in this case in the Jacobson radical, hence is $0$. 

The imposed condition, together with the fact that $x$ does not divide $e$, implies that $(e-x)=(e-ex)=e(1-x)$ divides $e$. In particular, $(1-x)$ divides $e$, and $e$ divides $x$, so $(1-x)$ divides $x$. It also divides $(1-x)$, so altogether $1-x$ divides $1=(1-x)+x$. Thus, $(1-x)$ is a unit. 

To finish the argument, note that $x$ in the above argument can be replaced by any multiple $x'=rx$. So we obtain that for all $r,$ $1-rx$ is a unit, i.e. $x \in J(R)=0$.