Congressional committee members rejected President Enrique Peña Nieto's same-sex marriage proposal; gay marriage is currently legal in 10 of Mexico's 32 states.
Medrano reassured his allegiance to his party, but he said that as a gay man he cannot vote against the gay marriage bill, which his party does not support.
According to the National Front, marriage is a union between one man and one woman for the procreation of children
The Front will also demand that certain content it deems to be "harmful" and "unsuitable" for children be removed from public education material.
Archdiocese compares government anti-discrimination agencies and the UN to the Gestapo, Nazi Germany's official secret police.
Supreme Court struck down laws passed in three states that define marriage as an act between one man and one woman for the purpose of procreation
Sunday's demonstrators carried a banner reading "We are families too" and placards reading "I respect your family, respect mine."
Before the voting held on Sunday, some priests urged their parishioners to reason their vote and to take into account the negative effects of gay marriage and adoption.
"Only the exclusive and indissoluble union between a man and a woman has a full social function," Mexico's Catholic Church said in a statement.
Rev. Hugo Valdemar, spokesman for the Catholic Archdiocese of Mexico City, urged lawmakers to listen to their "conscience" and oppose the initiative.