The equivalent statement of " If three vertices of a triangle are represented by cube roots of unity, then the triangle is an equilateral triangle " is

Let the statement be: If three vertices of a triangle are represented by cube roots of unity, then the triangle is an equilateral triangle.

In symbolic form, let $P$ represent three vertices of a triangle are represented by cube roots of unity and $Q$ represent the triangle is an equilateral triangle.
The given statement is $P \rightarrow Q$.

Logically, $P \rightarrow Q$ is equivalent to its contrapositive, $\neg Q \rightarrow \neg P$.

Analyzing the options:

Option A: $P \land \neg Q$ (the negation, not equivalent).

Option B: $Q \rightarrow P$ (the converse, not equivalent).

Option C: $\neg P \rightarrow \neg Q$ (the inverse, not equivalent).

Option D: $\neg Q \rightarrow \neg P$ (the contrapositive, equivalent to $P \rightarrow Q$).

Hence, the logically equivalent statement is option D.

The final answer is $\boxed{D}$