It's pretty easy - maybe 3 lines of code. First call bwdist to get the distance from the white pixels to the nearest black pixel. Then call bwmorph to skeletonize that so you get the distances along the centerlines. That's the radius so multiply by two to get the diameter. It will be an image looking like a skeleton and to find the diameter you simply look at the value of the pixel at the location you want the diameter of. Basically it's this (untested):
edtImage = 2 * bwdist(~binaryImage);
skeletonImage = bwmorph(binaryImage, 'skel', inf);
diameterImage = edtImage * double(skeletonImage);
Can't get much simpler, though you may want to add a utility where you let the user call ginput() to report back the diameter at the closest pixel to where they clicked. It's pretty easy, though clever and not obvious (until you know about it), so most novices might embark on some complicated looping process to get cross sections (what a nightmare that would be).