How can I tell if a closed path contains a given point?

The android.graphics.Path class doesn't have such a method. The Canvas class does have a clipping region that can be set to a path, there is no way to test it against a point. You might try Canvas.quickReject, testing against a single point rectangle (or a 1x1 Rect). I don't know if that would really check against the path or just the enclosing rectangle, though.

The Region class clearly only keeps track of the containing rectangle.

You might consider drawing each of your regions into an 8-bit alpha layer Bitmap with each Path filled in it's own 'color' value (make sure anti-aliasing is turned off in your Paint). This creates kind of a mask for each path filled with an index to the path that filled it. Then you could just use the pixel value as an index into your list of paths.

Bitmap lookup = Bitmap.createBitmap(width, height, Bitmap.Config.ALPHA_8);
//do this so that regions outside any path have a default
//path index of 255
lookup.eraseColor(0xFF000000);

Canvas canvas = new Canvas(lookup);
Paint paint = new Paint();

//these are defaults, you only need them if reusing a Paint
paint.setAntiAlias(false);
paint.setStyle(Paint.Style.FILL);

for(int i=0;i<paths.size();i++)
    {
    paint.setColor(i<<24); // use only alpha value for color 0xXX000000
    canvas.drawPath(paths.get(i), paint); 
    }

Then look up points,

int pathIndex = lookup.getPixel(x, y);
pathIndex >>>= 24;

Be sure to check for 255 (no path) if there are unfilled points.