If I am given the exact coordinates of three vertices in image-space (as in, the numbers are not integers, but can be rounded to find their location on the screen), how can I determine which pixels are inside the triangle and should be colored?

If you look at the diagram below it might clarify what I'm trying to do
As you can see, the pixels which are mostly within the triangle are filled green. How can I do this in code?

No need to code that youself. The Shape class has a Polygon subclass that allows you to create a Shape from an arbitrary list of x,y coordinates. The Shape class has methods for contains(Point2D p) etc, and the Graphics2D class knows how to paint/fill Shape objects.

Yeah the only thing was that I had to do per-pixel calculations