To evaluate each statement, let's analyze them one by one:

A: The square root of $25$ is $5$

- The square root of $25$ can be both positive and negative, so it is $\pm 5$. Therefore, statement A is incorrect because it does not account for the negative square root.

B: $9$ is a square root of $3$

- The square root of $3$ is not $9$. In fact, $9$ is the square of $3$. Therefore, statement B is incorrect.

C: The cube root of $0.8$ is $0.2$

- The cube root of $0.8$ is not $0.2$. To find the cube root of $0.8$, we are looking for a number that, when cubed, gives $0.8$. $0.2^3 = 0.008$, not $0.8$. Therefore, statement C is incorrect.

D: The square root of $(-2)^{2}$ is $2$

- First, calculate $(-2)^2 = 4$. Then, the square root of $4$ is $\pm 2$. However, when we talk about the principal square root (which is usually meant in these contexts), it is $2$. Therefore, statement D is correct because it refers to the principal square root.

Given these evaluations, the correct statement is:

$$\boxed{D}$$