However, there is the story of Radha and Krishna getting married by Brahma in the forest.
Radha is Krishna's energy/shakti. He cannot live without her, and she cannot live without him. She is not a woman, nor a jiva. She is a part of Krishna. She did not take birth. She emerged from him. Literally, they are one. You cannot have an image of Krishna alone. Krishna alone is incomplete without his Radha.
In some descriptions, radha was actually a married woman. it was she who is said to indulge in an affair with krishna. even if it was not so, the relationship of radha and krishna were on a different plane, and they were actually the same being, just god in the form of man and woman. their love was pure and spiritual, and cannot be purified further by the bond of marriage.
Radha and Krishna are said to be one. Their love is eternal. What is marriage? Marriage is a sacrament (Sanskara), a rite enabling two individuals to start their journey in life together. In a Hindu wedding, the multiplicity of creation becomes possible when spirit (Purush) unites with matter (Prakritti). Krishna and Radha being one did not need the basis of marriage.
This is of course my point of view, from what i have read and gathered. I myself am totally in love with Lord Krishna.