To determine whether point \( D(6, -3) \) lies on circle \( C \) with center at \( (6, 2) \) and a radius of 5, we need to calculate the length of segment \( CD \) and compare it to the radius.
The length of segment \( CD \) can be calculated using the distance formula:
\[ CD = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, the coordinates of the center \( C \) are \( (6, 2) \) and the coordinates of point \( D \) are \( (6, -3) \).
Substituting the coordinates into the distance formula:
\[ CD = \sqrt{(6 - 6)^2 + (-3 - 2)^2} = \sqrt{0^2 + (-5)^2} = \sqrt{25} = 5 \]
Since the length of segment \( CD \) is 5 and the radius of the circle is also 5, point \( D \) does indeed lie on circle \( C \).
Thus, segment \( CD \) must have a length of \( \boxed{5} \).