Just going by Wikipedia here:
"Marriage is a social union or legal contract between people that creates kinship. It is an institution in which interpersonal relationships, usually intimate and sexual, are acknowledged in a variety of ways, depending on the culture or subculture in which it is found. Such a union, often formalized via a wedding ceremony, may also be called matrimony."
"People marry for many reasons, including one or more of the following: legal, social, emotional, economical, spiritual, and religious."
The impression I get is that marriage plays a big part in a lot of religions. It is not, in and of its self, a wholly religious practice; nor has it ever been.
On the Ancient Greeks and marriage:
"In Ancient Greece, no specific civil ceremony was required for the creation of a marriage ? only mutual agreement and the fact that the couple must regard each other as husband and wife accordingly."
I can't see any mention of religion in the rest of the passage concerning Greece either (its under 'Marriage' on Wikipedia).
Likewise:
"Marriage, as we know it in our Western civilization today, has a long history with roots in several very different ancient cultures, of which the Roman, Hebrew, and Germanic are the most important. Western marriage has further been shaped by the doctrines and policies of the medieval Christian church, the demands of the Protestant Reformation, and the social impact of the Industrial Revolution."
"Further been shaped" - not created by.
Furthermore: "In ancient Greece marriage was seen as a fundamental social institution."
http://www2.hu-berlin.de/sexology/ATLAS_EN/html/history_of_marriage_in_western.html
From my understanding, religion is not a wholly religious practice, nor has it ever been, but that seems to be what you're basing your opinion on. I see no reason to change the word marriage then!