Ever participated in a wedding performed by a Justice of the Peace? Absolutely no religion there, Chris, yet it is every bit as legal and as binding a wedding as would be performed by your [insert favorite religious "officer" here].
In several states (including, ironically, Tejas), common-law marriages are considered legal and binding. Not even so much as a ceremony, yet the union must be broken up by a divorce decree, the partners share the responsibilities and privileges bestowed upon married people, etc.
"Holy Matrimony" is merely a term for the Dearly Beloved Gathered Here Before Me.... Marriage is a state of being imbued by the state, not a religious title. The fact that you can be married by a religious 'officer" is mererly a convenience borne (again) from Common Law.