Mexico's 2-1 win over Saudi Arabia was not enough to secure a round-of-16 spot as Poland finished second behind Argentina in Group C on goal difference.
After the game, former Barcelona boss Martino - who has been in charge of Mexico since January 2019 - said: "The contract expired with the final whistle and there is nothing else to do.
"I am the man responsible for the frustration and disappointment.
"It is real sadness and I assume all the responsibility for this huge failure."
Mexico opened their World Cup campaign with a goalless draw against Poland before losing 2-0 to Argentina.
Poland's 2-0 defeat to Argentina on Wednesday night gave Mexico an opportunity to climb into second place.
But, with all tie-breakers level before Poland had an advantage over Mexico with fewer yellow cards, El Tri knew they needed to add to the two-goal lead provided against Saudi Arabia by Henry Martin and Luis Chavez.
Mexico had two goals disallowed in the second half before Saudi Arabia secured Poland's progress via goal difference with a stoppage-time strike from Salem Al Dawsari.
Martino said: "We outplayed the opponent and created more chances and we could have scored as many goals as needed.
"That is why we went for the third goal and we added two centre-forwards together.
"We knew that Poland had fewer yellows and we were going out, so it was necessary in that moment.
"But we failed and I don't think we were knocked out in the previous matches, rather today.
"I think we should analyse this more thoroughly when we have a clear mind.
"This is the reality. We are in a position of fragility because we are out of the World Cup."
Saudi Arabia head coach Herve Renard said his players and staff would always remember a World Cup that saw them take the scalp of Argentina.
Renard said: "Afterwards, I said to them, 'congratulations for your World Cup, but we didn't deserve to win the game'.
"We won't forget what we did together at this World Cup. This was more difficult for us, but I will always protect them and I'm proud of their World Cup."
Victory would have seen Saudi Arabia progress from the group stage for only the second time.
Renard said: "The World Cup standard is always high and in our group we had three very good teams.
"In Saudi Arabian history we have only once reached the knockout stages and that was a long time ago, back in 1994.
"We were dreaming of doing it again, but the way we played here we didn't deserve to do that."