Evgeny's answer works, but it's a lot of effort without library support: compute a full Voronoi diagram plus an additional sweep line algorithm. It's easier to enumerate for both sets of points the points whose Voronoi cells intersect the separating line, in order, and then test all pairs of points whose cells intersect via a linear-time merge step.
To compute the needed fragment of the Voronoi diagram, assume that the x-axis is the separating line. Sort the points in the set by x-coordinate, discarding points with larger y than some other point with equal x. Begin scanning the points in order of x-coordinate, pushing them onto a stack. Between pushes, if the stack has at least three points, say p, q, r, with r most recently pushed, test whether the line bisecting pq intersects the separating line after the line bisecting qr. If so, discard q, and repeat the test with the new top three. Crude ASCII art:
Case 1: retain q
------1-2-------------- separating line
/ |
p / |
\ |
q-------r
Case 2: discard q
--2---1---------------- separating line
\ /
p X r
\ /
q