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Yes Ganga is married to Shiva and the Devi Bhagvatam Book 9 Chapter 6 gives the reason:

16-21. Nârâyana said :-- Hear, O Nârada! I will now describe that incident, the hearing of which removes all the sins. Laksmî, Sarasvatî and Gangâ, the three wives of Hari and all equally loved, remain always close to Hari. One day Gangâ cast side-long glances frequently towards Nârâyana and was eagerly looking at Him, with smile on Her lips. Seeing this, the Lord Nârâyana, startled and looked at Gangâ and smiled also. Laksmî saw that, but she did not take any offence. But Sarasvatî became very angry. Padmâ (Laksmî) who was of Sattva Guna, began to console in various ways the wrathful Sarasvatî; but she could not be appeased by any means. Rather Her face became red out of anger; she began to tremble out of her feelings (passion); Her lips quivered; and She began to speak to Her husband.

Lord Vishnu perhaps realizing that he should best remain out of a the women's argument left the place and then started the curses and counter curses. Saraswati tries to attack Ganga but Lakshmi intervenes so gets cursed by the former. Seeing this Ganga gets angry and curses Saraswati:

39-44. Thus saying, Gangâ became ready to curse Sarasvatî and addressing Laksmî, said :-- “O Dear Padme! As that woman has cursed you to become a river, so I too curse her, that she, too, be turned into a river and she would go to the abode of men, the sinners, to the world and take their heaps of sins.” Hearing this curse of Gangâ, Sarasvatî gave her curse, “You, too, will have to descend into the Bhurloka (the world) as a river, taking all the sins of the sinners.”

When all the action is over Lord Vishnu returns and tells them:

O Gange! You will also have to take incarnation in Bhârata as a river, purifying all the worlds, to destroy the sins of the inhabitants of Bhârata. Bhagiratha will take you there after much entreating and worshipping you; and you will be famous by the name Bhagirathî, the most sanctifying river in the world. There, the Ocean born of my parts, and the King S'ântanu also born of my parts will be your husbands.

O Bharatî! Let you go also and incarnate in part in Bhârata under the curse of Gangâ. O Good-natured One! Now go in full Amsas to Brahmâ and become His wife. Let Gangâ go also in Her fullness to S'iva. Let Padmâ remain with Me. Padmâ is of a peaceful nature, void of anger, devoted to Me and of a Sâttvika nature.

Therefore, Ganga came to be Shiva's wife as a result of the events that were set in motion by Sarwaswati's curse.