As Peter mentions, in general English has no distinction between male and female with verbs relating to marriage or relationships. For example, "married", "dating", "seeing each other", "in a relationship", "partners", "having a fling", "friends with benefits", "casual sex", "one-night stand", and many others, all are gender-neutral and can apply equally to straight or gay couples.
However there are many words to characterize the nuances of married life, many of which are colloquial or vulgar. Aside from the basic honorifics (Mr., Mrs. Miss, and Ms.) there are these:
A "henpicked" husband is one whose wife constantly nags him and orders him around, and who is generally the subordinate in the relationship.
A "matriarch" is a strong woman who is the head of a (usually large) family, and who is usually controlling. A "patriarch" is similar for men, although there can be some religious overtones, as with the male leader of a religious group.
For gay male couples, there is often some distinction between the "top" and the "bottom" in the relationship.. More crudely this can be referred to using baseball metaphors as "pitcher" and "catcher".
A man who supports a (often much younger) woman financially, usually as his mistress, would be called a "sugar daddy". A woman who is in a relationship simply because the other person is wealthy would be called a "gold-digger". Note these terms can also be used for the opposite gender ("sugar momma/mommy"), or for gay relationships.
As far as I know there is no special term for a homosexual marriage. We just call it a "marriage". Since it's not always clear how to refer to the two people in the marriage, "partner" is a good catch-all term instead of "husband" or "wife". Note "partner" can also be used for heterosexual relationships, married or otherwise, so it can be somewhat ambiguous.