You almost have it. *Really* close.

Formula for line from (-2, 3) with slope -4/3 is

y - 3 = -4/3(x + 2)

Now find where this intersects

4y - 3x + 2 = 0

These should intersect at one point, P, because they are perpendicular.

Calculate the distance from (-2, 3) to P, and this is the radius for your circle.

This touches the line once. Why? The radius from (-2,3) to P forms right angles with the line.