This will give an array mapping the distance from every object to every other object. You could reduce this with triu() if you want, due to the symmetry.
inpict = rgb2gray(imread('dots.jpeg'))>128;
L = bwlabel(inpict); % this identfies all the objects
C = regionprops(inpict,'centroid');
C = vertcat(C.Centroid);
D = sqrt((C(:,1)-C(:,1).').^2 + (C(:,2)-C(:,2).').^2);
If you wanted to find the distance to the nearest object, you could use this (there are probably other ways).
D(abs(D)<1E-6) = NaN; % remove zeros
[Dn Nn] = min(D,[],2); % minimize
% Dn is distance to nearest neighbor
% Nn is nearest neighbor
.
