The comments, which were made last weekend, were met with a backlash in Mexico.
President Claudia Sheinbaum, the country's first female president, said: "Chicharito is a very good soccer [player] but when it comes to his opinion on women... he still has a lot to learn."
In a joint statement, the FMF and Mexico's top men's and women's leagues said the Gender and Diversity Commission has started an investigation "that seeks to prevent and punish this behaviour" and it will "take more severe measures in the event of a repeat offence".
Chivas said the comments were "contrary to the principles and values" of the club and that they had taken "appropriate action" against Hernandez, though they did not specify which action.
Hernandez, Mexico's all-time leading scorer, joined Manchester United from Chivas in 2010.
He spent four years at Old Trafford, winning the Premier League twice.
After spells with Real Madrid and Bayer Leverkusen, he returned to the Premier League with West Ham for a three-season stint between 2017 and 2020.
Hernandez then joined Sevilla from the Hammers, before spending four seasons with MLS side Los Angeles Galaxy.
He returned to former club Chivas in 2023, signing a two-year deal.
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