"I'm the husband of the wife you wrote to," Rodriguez wrote in Spanish on Instagram. "I understand that my work in the national team can be the cause of all of this in you. I don't justify it, but I understand it.
"I ask you please not to write anymore to my wife, to rape and kill are very cruel words. We don't deserve this, least of all to include family. I hope this is over now. May God bless you."
The threatening messages were posted after Rodriguez started in Mexico's 2-0 loss against the United States in Friday's World Cup qualifier in Cincinnati.
The outcome saw Mexico lose top spot in the CONCACAF group on goal difference, with the two teams now level on 14 points after seven games.
With the threats continuing, Rodriguez decided to share the disturbing messages his wife received.
"I honestly take into account the positives and negatives..." Rodriguez wrote. "Football is and will always be a dream come true for me. To leave the national team is not an option for me."