One argument used to justify the expansion of marriage to include same-sex couples is the example of male and female couples who are infertile.  The ‘Discussing Marriage‘ website features a somewhat foundational philosophy of marriage called the conjugal view of marriage which I may cover later.  But in a nutshell, this view forwards the idea that traditional marriage is a unique relationship which the state has an interest in because of the potential for procreation.  Other relationships may be valuable and fulfilling, but since there is not the potential for procreation in these relationships, the state does not have the same type of interest in them.  Thus no need to treat these other relationships in the same way they would treat a traditional marriage.

So the argument to justify the expanding definition of marriage to include same-sex couples here would be:

P1:  Some traditional couples are married, yet cannot have children
P2:  Traditional couples that cannot have children can continue in a marriage relationship recognized by the state
P3:  There is no fertility requirement for marriage
C1:  The marriage definition should be expanded to include same-sex couples

Responses to this argument are many, and I would refer you to the link above to review many responses.  The main responses to me is that infertility is often unknown prior to marriage, and infertility is sometimes treatable, and may be cured at some future time.  So P1 is often not known going into a marriage, and may not be a permanent condition.

P2 is understandable because infertile male-female marriages do not undermine the conjugal view of marriage the way other relationships would if they were called marriage as well.

P3 is a statement that leads to what Discussing Marriage calls the revisionist view of marriage.  In this view the unique unifying good of marriage is more along the lines of personal fulfillment rather than potential for procreation.  Which does not have as compelling a reason for state involvement.