20 people were killed in the shooting at a Walmart in El Paso, Texas on Saturday, and dozens more were injured.
Ebrard said Mexico condemned Saturday's shooting and called it an "act of barbarism".
During a press conference in Mexico City, Ebrard called the shooting an act of terrorism against Mexicans in the us and said the Mexican government will look into whether there is enough evidence to solicit the extradition of the gunman to face charges in Mexico.
The President said in his final proposal that he would be directing the Department of Justice to propose legislation that would ensure "those who commit hate crimes and mass murders face the death penalty, and that this capital punishment is delivered quickly, decisively, and without years of needless delay".
He said his government was studying the possibility of accusing the suspected shooter, named in United States media as 21-year-old Patrick Crusius, of "terrorism" and requesting his extradition to face charges in Mexico. The average Mexican has trouble legally acquiring a handgun, much less an assault weapon.
Another mass shooting took place early Sunday at a bar in Dayton, a city in the state of OH, killing nine people and injuring 26 others.
Mexican Foreign Minister Marcelo Ebrard said the country will take action under worldwide law. It also wants to extradite Crusius to Mexico.
"If we look at things objectively, we have to say that the two main USA parties have given little attention to gun control", Mr Lopez Obrador said in his morning press conference.
Ebrard also said Mexican President Andres Manuel Lopez Obrador has ordered prompt and forceful legal action to protect Mexican-Americans and Mexicans who are visiting the U.S.
Major Mexican dailies widely reflected the Mexican government position, expressed in a tweet that said: "We condemn this act of terrorism against the Mexican-US community and nationals from Mexico in the United States of America".
"What happened is inadmissible and today, at 4 p.m., we will announce the first judicial actions that the Mexican government will take in accordance to global law", he added.
Investigators believe the shooter uploaded a rambling screed online shortly before targeting a busy shopping area, railing against a perceived "invasion" of Latin Americans coming into the US.
In an editorial entitled "Elements for hatred", El Universal described the El Paso killings as "the biggest massacre of persons of Mexican origin in the United States".
"For Mexico, this individual is a terrorist", he said.
El Paso sits on the border across from the Mexican city of Juarez. At least 26 people were wounded.