When analyzing the relative sizes of $2$ and $-|-2|$, the expression containing the absolute value is prioritized during simplification. The absolute value of $-2$ is $2$, considering its distance from zero. Enacting the negative sign placed outside these absolute value bars negates this distance, yielding $-2$. The evaluation then compares the initial positive integer $2$ against this resulting negative integer $-2$. Because any positive integer is strictly greater than a negative one ($2 > -2$), the final comparative relationship is $2 > -|-2|$.